MENSURATION OF TIMBER

CARTER,

LIBRARY

FACULTY OF FORESTRY
UNIVERSITY OF TORONTO

A
TREATISE

OK TUB

MENSURATION OF TIMBER

AMD

TIMBER CROPS.

P. J. CARTER,

FOHBST DKPAHTUBXT.

from the <Dffi« of the Inspector ©eneral of ^forests,

f/»*r

b>

fiction.

CALCUTTA :

OFFICE OF THE 8UPEBINTENDENT OF GOVERNMENT PRINTING, INDIA.

1893.

Price Eight Annas.

INTRODUCTION.

treatise appeared originally in the Indian
Forester, and it was there explained that it was for
the most part a compilation from various German text-
books, special use having been made of the article by
A. Ritter von Guttenberg in Lorey's "Handbook of
Forestry " published in 1887. In the first instance
the German originals were closely followed and special
regard was paid to diameter measurements, which give
more nearly the actual contents of logs and trees, and
which are used on the continent of Europe by buyers
and sellers of timber ; but, since in England and India
the contents of round timber are always calculated on
the assumption that the sectional area of a log is deduced
from the square of the quarter-girth, it has been thought
advisable in the present edition to have regard to the
established custom, and thus render this treatise of
more practical value to the English and Indian student.

P.-J. CARTER.
June 18V2.

5O
-555

CV7

TABLE OF CONTENTS.

The division, and grouping of the subjects, treated of in the
following pages, is as follows : —

PAGB
CHAPTER I. — The measurements of heights, lengths, sectional areas,

girths, and diameters of stems and trees . . 1

„ II. — The measurement of felled trees .... 4

y III. — The measurement of standing trees ... 7
„ IV. — The measurement or valuation survey of standing crops 10

., V. — The determination of the ages of trees and crops . 37
„ VI. — The determination of the rate of increase of individual

trees, and of crops of trees .... 43

M VII. — The compilation of tables of yield •'..** 65

2

catch the plummet line and keep it in place until the figure indi-
cated has been read.

For measuring lengths, graduated rules or tapes may be 'used.
Where great accuracy is required, the length of a felled tree or log
should be measured along, or parallel to, its axis, and not on its
plopping surface.

The sectional area of a log or tree can very rarely indeed be
obtained directly. In nearly every case the girth or diameter must
be measured, and the area of the section determined as if the sec-
tion were a circle. Area = " ' - = — (diameter)8 or according

47T 4

/girthV

to trade custom = I2- — I
\ 4 /•

Girths are measured with tapes. It is convenient to have tapes
graduated on both sides — one side for reading the girth, and the
other for reading the corresponding diameter. The zero end of
the tape should be furnished with a sharp metal point that can be
easily fixed in the bark of the tree, so that one person may be able
to measure any stem, no matter how thick it is. As a circle en-
closes a greater area than any other plane figure of equal perime-
ter, and as the sectional outline of trees is seldom quite circular*
the contents of a log or tree calculated directly from its girth by

the formula " - will usually be in excess of the true contents.
4r

Unless the contour of the log is circular, it is impossible to obtain
by girth measurement the circumference of the circle which encloses
the same space as the section whose area is required. Irregularities
of outline, due to fluting, bark, etc., cannot be overcome in measure-
ments of girth, whereas, as^we shall presently see, they can more or v
Jew successfully be allowed for in measuring diameters. Experi-
ments made in Baden prove that girth measurement yields a result
that is from six to ten percent, greater than that obtained by means
of diameter measurement. It is, however, obvious that in cubing
log* departing from the cylindrical form, the measurement of the
i;irth is more to be relied on than the measurement of a single
diameter.

When the true contents of a log are to be deduced from diame-
ter measurement, that diameter shouFd be sought which, considered
nut the diameter of a circle, gives a result as nearly as practicable

equal to the area of the section measured. When the section is
elliptical, the mean of the longest and shortest diameters should
be taken | — ±_) and the area of the section is then assumed to be

»r
- x

2
2Vd

Now as the real area of the ellipse is -

*

the mode of measurement recommended gives an excess of
\ (—4-) , tnat " to say» an excess equal to the area of a circle
whose diameter is equal to half the difference of the two measured
diameters. Save in very exceptional cases, this difference is small
enough to be disregarded. The area of sections of irregular contour
can be determined from the mean of three diameters, but the
result thus obtained will generally be found to be somewhat too
high.

Diameters are measured with a calliper resembling, in all its essen-
tial parts, a shoemaker's measure. The pattern of calliper invented
by Friedrich is one of the best. It consists of a graduated rule
A A, to one end of which is fixed the arm BB. CC is a moveable
arm capable of sliding backwards and forwards on the rule A A
which passes through the hole a b cd in it. To enable the arm to
slide freely the hole is made oblique to its inner face, but iu such
a> manner that as soon as it comes -in contact with the log to be
pmbrnoed, the arm is pushed back and rests perpendicularly on
the graduated rule, with which it remains in contact only along the
edges b and c.

A si

In men?uring logs and trees the following general rules are laid
down for foresters on the continent of Europe :—
(a) Diameters are to be preferred to girths.
(l>) In thp ca«e of elliptical or oval stems, take the mean of the

largest and smallest diameters.
(e) In the case of large stems measure at least two diameters.

(d) In the case of stems of irregular contour, measure several

diameters, and avoid all protuberances, etc.

(e) Measure diameters and girths always in a plane at right

angles to the axis of the stem.

(/) If the place of measurement falls on an irregular part of
the stem, measure the diameter or girth, as the case
may be, at an equal distance on either side (above and
below) of the irregularity, and take the mean of the two
measurements.
(y) Moss, etc., thick enough to vitiate the measurement of the

stem should be removed.

(A) If an accurate measurement of an irregular section is re-
quired, transfer its outline to tracing paper and compute
its area with a planimeter or acre-comb.
(i) Never be without tables showing at a glauce the areas of

circles for given diameters and girths.

In England and India diameters are rarely measured, as old
established custom has prescribed that in the sale of timber the
square of the quarter-girth should be regarded as the sectional area
of the log or tree.

Chapter II.

On the measurement of Felted, Trees.

The felled trees should be cut up in the usual way, that is to
say, into logs and smaller pieces.

1. 3Jfat*rcme>j,f. 'of linuml Timber.

Several formula have been devised for the determination of
the contents of round tin-ber with more or less near approach to
accuracy, but only two are ' of practical utility. These are,

(i) — ' x I ......... known' as Smalian's formula; and

(ii) omt ............. knctfu as I tuber's formula.

5

In the above / is the length of the log, aby am and at the sec-
tional area of the log at the base, middle and top, respectively.

Both formulae contain an error, the extent of which is propor-
tionate to the amount of difference between the diameters at the
top and base, respectively, of the log, that is to say, to its degree of
taper; and this error increases as the square of that difference.
Huber's formula always gives too small, and Smalian's too great, a
result, the error of defect in the one case being one-half the error
of excess in the other.

Huber's formula has also another advantage, for which it ia to
be preferred : the modes of measurement and calculation adopted
in France and Germany give, as a rule, too high a figure for the
sectional area concerned in each case. This excess is partly com-
pensated for by the employment of Huber's formula, whereas the
other would only exaggerate it.

In order still further to diminish error, long logs should be mea-
sured in two or more sections, the number of the sections increas-
ing, i.e., their length diminishing, with the taper of each log.
The contents of those of regular shape and not exceeding 20 feet in
length may, however, be deduced from their sectional area in the
middle. Longer logs, even if of regular shape, should be cubed
in 'two or three sections.

All large round logs should be measured singly.

If the logs are stacked so that they cannot be conveniently
measured in the middle, the mean of the sectional areas at the base
and at the top must be taken. The mean sectional area should
never, under any circumstances, be deduced from the mean of the
two girths at the two extremities respectively, or an error of
from 10 to 15 per cent, may result.

Poles are seldom cubed singly; nearly always in stacks, built
up of poles of one and the same length, and of approximately one
and the same girth. Their solid contents are generally ascertained
by inspection from special tables.

Straight and regular-shaped branches are measured in the same
way as logs.

2. Measurement of Square-cut Tiinber.

Such timber must of course be cubed by the formula, length
x width x thickness.

6

3. Measurement of Small Wood.

The solid contents of toppings and lopping-?, and of irregular-
shaped pieces from stumps and roots, are obtained by the water
method (being equal to the quantity of water they displace when
submerged) or by the water-method and weighment combined.
For the water-method special vessels, called xylometert, may be
employed. In the combined system samples of each kind or class
of wood are successively weighed and measured by the water-
method, and the contents of the entire quantity in each class are
then worked out by means of a simple proportion sum. Figures
expressing specific gravity cannot be employed, since the specific
gravity of wood varies not only according to the amount of mois-
ture present, but even in one and the same tree according to the
part from which it is derived.

The most rapid way of measuring small wood on a large scale is
to stack it cut up into billets of one and the same length, the
width of each stack being equal to the common length of the
billets. The contents of a stack will be equal to length x height
x common length of the billets. The length of a stack built up
on a slope must be measured horizontally. The above formula will
give us only stacked contents ; to reduce these to solid contents, we
must determine, by the water-method, or by the combined water
and weighment method, the exact volume of a sufficiently large
number of stacked units, thereby obtaining the ratio between solid
contents and stacked contents. To obtain the solid contents of a
stack we have then only to multiply the stacked contents by this
ratio, which we may hence term a reducing factor. The following
figures may be accepted asiaverage reducing factors for converting^
•tacked into solid contents :—

For split wood . * . . 0-60 to 0-80

„ round billets . .. .' . 0'50 „ 0'65

„ email stuff . ' . . . O'SO „ 0-45

„ wood from stumps and roots . 0*30 „ 0'40

In connection with the determinaiion of the solid contents of

wood it is obvious—

(a) That the longer the billets are, or the less carefully built
up the stacks are, the less will be the solid contents.
lu can-less stacking billets often lie across one another.

(6) That the thicker or more regular-shaped the billets are,
or the more carefully built up the stacks are, the
greater will be the solid contents.

(c) That the larger the stacks are, the larger will be the re-
ducing factor to be adopted.

4. Measurement <>f Bark.

"When bark is sold separately, its quantity may be determined
•either by weighment or by ascertainment of volume. The solid
contents are calculated by means of reducing factors in the same
way as the solid contents of femall wood. Experiments give from
0'3 to 0*4 as the average factors for bark. It has been found that
the quantity of bark varies from 6 to 15 per cent, of the total
volume of the tree or crop.

Chapter III.

On the measurement of Standing Trees.

In this case, unless ladders are used (a procedure that is hardly
practical and is not really necessary) only a siny;le diameter or
gi-rth can be measured directly, viz., near the base of the tree.
Any diameter above 6 feet from the ground must be measured
indirectly by means of special instruments, the best of which are
Winkler's and Saulaville's Dendrometers and Breymann's Univer-
sal Instrument. Obviously no direct measurement of the branches
is practicable, and their cubical contents can, therefore, only he
estimated from the results of special experiments, or with the help
of long experience.

We have five different methods of estimating the contents of
standing trees—

1. Ocular estimation, without any measurement at all.

£. Estimation with the help of mass-tables, the height and

girth at breast-height being accurately measured.
3. Estimation with the help of farm-factors, which serve to
-. , reduce, to the true contents of the tree or of any part of
the tree, the volume of the cylinder, whose height \B the
height of the tree and girth the girth of the tree
measured at breast-height.

8

4. Estimation by r»c£/-height, in which, besides the girth

at breast-height, the height at which the stem tapers
down to half that girth is measured.

5. Estimation with the help of the height of the tree and

several girths, the lowest of which is measured at breast-
height.

1. Ocular Estimation.

Practised wood-cutters are able to estimate more or It ss accur-.
ately with the eye alone the contents of trees belonging to species
that they are familiar with, and growing in localities with the
peculiarities of which they are acquainted. It is needless to say
that the most experienced are liable to commit large errors, and
that the inexperienced should never employ this method.

2. Estimation with the help of Mass-tables.

The mass-tables drawn up with great labour for the forests of
the kingdom of Bavaria give the cubical contents of trees of known
height, diameter, and age-class. They comprise averages deduced
from the measurements of 40,000 trees. On this account, although
they give accurate results for a large number of trees taken to-
gether, they are not to be relied on for cubing trees singly, as the
single tree in any given case may ditfer very widely from the
average tree.

3. Estimation by weans of Form- factors.

If a = sectional area of the trunk at breast-height, h = height
of the tree, c = the true contents of the tree or tree-part consider-
ed, and C = volume of an ideal cylinder whose basal area is a and
height kt then we have the following formulae : —

rf

where/ is a constant termed the form-factor, and is deduced as an
average from the measurement of a sufficiently large number of
type trees. Type trees are selected, felled and measured separately
for each age or si/te-class, and for each species or -group of species.

Form-factors may be deduced, according to the requirements of
the case, for the stem only, or for the whole tree, or for the timber

9

only, or for the branches, or for the roots, or for all and each
severally.

In the formulae above we have supposed that the girth and
sectional measurements have been taken at the height of a man's
chest above the ground, assumed, for the sake' of uniformity, to be
4 feet 3 inches. But it is obvious that any other conventional
height would serve the purpose, although it is usual and most
convenient to employ the one we have adopted. We need refer to
only one other convention which is sometimes used. The girth
may be measured at a constant fraction (say, for instance, one-
twentieth) of the height of the tree, in which case the form-factors
obtained are termed normal. Normal form-factors yield perfectly
correct results, but they are not practical owing to the difficulty
and trouble of measuring at such various heights, many of which
cannot be conveniently reached.

Form-factors are said to be absolute when the base of the ideal
cylinder is assumed to be in the same plane as the girth which
is measured. In this case the contents of the portion of the stem
below the plane must be calculated separately.

Like mass-tables, form-factors give closer results for an entire
forest than for individual trees.

'The preparation of a complete set of form-factors requires great
care and experience, as their correctness depends entirely on the
selection of the type trees, whose dimensions serve as the basis of
all the calculations. In some cases the trees of a crop have been
classified into various classes according to their height and shape,
and a separate form-factor calculated for each class. The most
recent investigations prove that form-factors vary chiefly with the
height of the trees.

4. Estimation by r\c\\i-height.

By the term n'cM-height we mean that height at which the
stem of the tree measured has a diameter equal to half the diameter
at some point near the ground. If hr = the m-M-height hg =the
height at which the diameter near the ground is measured, and
a— the sectional area of the stem at this height, then, according
to Pressler, the contents of the stem

as ij ahr + ahg.

This formula is based on the fact that the first term represents

10

correctly the volume both of the cone and of the paraboloid, and is
only 1*8 per cent, less than that of a cone with a concave surface.

The rtf^-height may be estimated with the eye, or obtained
with the help of a special measurer .(the richt-tube). This instru-
ment consists of cardboard tubes telescoping one into the other.
At the objective end of the outer tube are fixed two wire points at
the extremities of one and the same diameter. The end of the
innermost tube is closed, except for a small hole to which the eye
is applied. To use the instrument, direct it on the trunk at the
point where the diameter has been measured, and pull out the
tubes until the wire points just embrace it. Then drawing- out the
tubes to twice this length, direct the instrument again on the tretj,
working it up along the trunk until the wire points just embrace it,
and note the point where this occurs. The diameter at that point
is, on the principle of similar triangles, half the original measured
diameter, and the height of the point is the rze/itf-height sought.

5. Estimation by means of several diameters.
The measurement of diameters above the reach of a man of
average stature requires the use of special instruments, and hence
this method is seldom employed.

6. General.

As the fourth and fifth methods can give only the contents of
the stem, the contents of the branches and stump and roots must
be obtained by means of special tables compiled for the purpose.

Chapter iy. ' ,
On the measurement or Valuation Survey of Standing Crops.

1. The Various Methods of Valuation Survey in General.
The most correct method of obtaining the cubical contents of
an entire crop would be to cube each component tree separately,
and then sum up the results. But such extremely detailed pro-
cedure is entirely impracticabie, except in the case of crops of very
limited extent. In practice, therefore, it is necessary to devise
some much more expeditious methods that will yield results accu-
rate enough for the purposes of the forester.

11

Without being guilty of any important error, we may assume
that in one and the same crop trees of like girth and height
do not differ greatly either as respects cubical contents or form-
factor. Hence, we may divide the component trees of a crop into
classes based on equality of diameter and height combined, and
find the contents of each class by selecting trees fairly representa-
tive of that class (sample or type trees) and measuring these sepa-
rately. The average contents of the sample trees, multiplied by
the number of trees composing the class, will give very approxi-
mately the true contents of the whole class.

If all the component trees of a crop were of like height, dia-
meter, aud form, the measurement of a single sample stem would
suffice. In reality, however, the trees of a crop are of unequal
development, and must be divided into classes comprising indivi-
duals of equal diameter and height. Nevertheless, it may some-
times be possible in a more or less irregular crop to find a tree
such that its contents are equal to the mean contents of all the
trees comprising the crop. Such a tree we may term an average
tree, and the method of measurement in which average trees are
employed may accordingly be termed Valuation by average treet.
Sjuppose c = the contents of the average tree, C, = the contents of
the whole crop, and n = the total number of trees, then —

It is, however, rare to find a single tree such that its contents are
equal to the mean contents of all the trees of the crop; while, on
the other hand, the establishment of as many classes as there are
different girths and heights present in the forest would involve
enormous expenditure of time and labour. Hence the adoption of
a middle course is to be recommended. Firstly, more compre-
hensive girth-and-height classes should be established, each class
comprising trees not precisely of one and the same dimensions,
but of different heights and girths varying between a maximum
and a minimum that are sufficiently close together to ensure the
necessary degree of accuracy-; and, secondly, these girth-and-height
classes being established, the average tree for each class should be
obtained by calculation. For convenience' sake we may designate
this method of measuring crops, Valuation by meant of girth. a*d-
height gradations.

12

Just as the contents of a crop or of a class muy be taken as the
product of the number of stems and the contents of the average
tree, BO we may also express it either as the sum of the basal areas
of all the stems (A) x the average -height of the crop (H) X the
average form-factor (F) or as A x the average rich ^-height (3r).

We have thus three formulae —

(i), C = e n ... Valuation by means of the average tree.
(ii), C = A HF.,. „ „ average form-factor.

(iii), C= \ A HT... „ „ average rtcAl-height.

For the first formula the sample stem measured must be such

Q

that its contents = —

n

In using the second formula the height and form-factor of the
sample stems measured must be identical with the average height
and form-factor of the crop or height-and-girth class, a condi-
tion that is more easily realized than the equality demanded for
the use of the first formula.

Of the quantities A, U, and P, the first is obtained at once by
direct measurement of the girths of all the stems taken at a
fixed height above the ground ; while the other two are ideal, and
must be obtained as accurately as possible by computation from
the measurements of the sample stems.

In those methods of valuation survey which are based on the
actually measured contents of sample trees and on the measured
basal areas of all the trees, a fourth formula, derived from formula
(ii), may be substituted for formula (i). Using the same expres-
sions as before, and supposing that (?, a, A, and/, respectively are
the mean contents, basal area, height, and form-factor of the type
or sample trees, we have — ' ,

£*= A HF. ................................ '. ........ Formula (ii),

Bndf = a x A x/. ............ „; ........ ........ by assumption ;

Hence C : c = A HF: ak f

and a* HF'ts by assumption = A /

....................... . ....... ....... Formula (iv).

reasons

Formula (iv) is to be preferred to formula (i), for two .

itly, because the important fcnd easily obtained term A enters

> >t; and, secondly, because the sample stems have to furnish
only the average height and form-factor, not the average contents,
of all the trees of the crop or of the diamoter-and-height class, so

13

that they need uot be average trees in the strict sense of the won).
After the measurements required to obtain the total basal areas
of the trees have been taken, it only remains for the surveyor to
select his sample trees properly and in suitable numbers for each
girth-height class. My suitable numbers is meant a fixed pro-
portion of the total number of trees in the respective classes, or
equal numbers in case the several classes include more or less the
same number of individuals.

The sample stems are usually felled in order to determine their
contents ; but their contents may be" obtained without felling by
means of volume-tables or of previously prepared tables of form-
factors.

The methods of valuation hitherto described require that every
tree in the forest should be measured. But the contents of the
whole crop may also be calculated by means of a sum of simple
proportion, from data furnished by sample plots in which alone
the trees are measured.

But measurement of every kind may even be entirely dispensed,
giving place either to ocular estimation, or to estimation by com-
parison with tesults obtained in similar crops elsewhere.

The following is a synoptical view of the various methods of
effecting a valuation survey of a crop : —

I. VALUATION BY ACTUAL MEASUREMENT.

A. Of every tree in the entire crop (complete survey).

B. Of every tree only in sample plots (survey by sam-

ple plots),

Whether we undertake a complete survey, or only
one by sample plots, we may ascertain the contents
of the whole crop —

a By deducing it from the contents of type or
sample trees, representing either —

a. The ideal average tree of the crop, or
/3. The -average of trees of one and the

same girth and height, or
•y. The average of trees of comprehen-
sive girth and height-classes (girth
and height varying between a maxi-
mum and a minimum limit), i.e., girtk
and height gradations.

14

Now whether we adopt method a, |3 or y, we
may obtain the contents of the sample trees
either (1) by felling them and measuring
them accurately, or (2) by estimating their
contents standing. lu either case, we may
seek to ascertain one of two things : (i) the
total solid contents of the trees, or (ii) sepa-
rately the quantity of each class of wood
or timber in them.

b. By means of the ric^-height.

c. With the help of specially prepared tables of

volumes (volume-tables) or of form-factors.

II. VALUATION WITHOUT ANY MEASUREMENTS (eye survey).

A. Ocular estimate, after observation either (a) of the

whole crop, or (b) of sample plots.

a. Of the number of stems of different size-classes.

b. Of volume of material (1) per acre, or (2)

standing in the whole forest.

B. Estimate based on examination of figures given in

existing yield-tables prepared either —

a. Specially for the locality, or

b. For the forest district or region.

2. Choice between Complete Survey and Survey by Sample Plots.

The valuation survey of a crop by means of sample plots obvious-
ly requires very much less labour and time than a complete survey,
and must therefore be adopted whenever it is likely to fulfil the
objects of the survey. Its admissibility depends on three principal
considerations : —

I. — THK PUKPOSK OF THK SURVEY AND THB DEGREE OF ACCURACY
DKUANDKD — The object of a survey is not necessarily always
to ascertain the total contents oi the crop : we may desire
to know only how much material on an average there is on
an acre, or we may seek to obtain figures required for the com-
pilation of certain tables, or we may simply wish to determine
the quality of the soil br locality, and so on. In all these
latter cases the survey of well-s'elected plots, the area of which
has been accurately measured, is preferable to the survey of

15

the entire crop, which will rarely be found to be of sufficiently
uniform quality and composition throughout. Moreover, the
area of a crop is often not exactly known. "When great accur-
acy is required, as when the whole of a standing crop is to be
put up to sale, it is of course advisable -to measure at least
the girth of every tree in the crop. Still it must be un-
derstood that, in the most carefully organized and conducted
valuation survey, only a limited degree of accuracy is attain-
able, for Although the girths or basal sections of the trees
may be obtained witli sufficient exactitude, the heights and
form-factors of the trees can only be determined approximate-
ly. For carefully-framed working plans it is usual to make
complete surveys, the procedure by sample areas being adopted
only when circumstances render a complete survey difficult
and at the same permit of sufficiently correct generalizations
from the part to the whole.

II. — THE SIZB AND NATURE OP THE CHOP. — Valuation survey by
sample plots is obviously admissible only in crops that are so
far uniform as to render it practicable to select certain portions
presenting the average characteristics of the whole ; but this
method of survey is not justifiable if no time is thereby saved.
Thus, if a crop is of limited extent, the whole of it can often
be surveyed as quickly as a sample plot, which has to be
carefully selected and then marked out and measured.. So also
in very open crops a complete survey is preferable, as it can
be effected rapidly, and the sample plot, to represent the aver-
age of such a crop, must be comparatively large. We may
lay down the following two rules for general guidance : —

1. In three cases the system of sample plots should be avoid-
ed— Firstly, in irregular crops of very variable density, or
containing trees of very different girths in their different
parts; secondly, in small crops not exceeding five acres in
extent; and, thirdly, in very open crops, or in crops in
which only certain scattered trees, such as coppice stores,
large trees in an area under jardinage, have to be accounted

i ' for.

2. On the other hand in young crops or in coppice, where
often 2,000 and even more stems may stand on an acre*

16

a complete survey is quite out of the question, aud sample
plots should be surveyed, if there are no volume-tables avail-
able to furnish the requisite data and dispense with the
necessity of any measurements.

III. — IN A CERTAIN SENSE, ALSO THE NATURE OP THE GROUND.

—On gentle slopes the whole crop can be easily surveyed,
but on steep or rough , rocky hill-sides, a complete survey
would be difficult as well as expensive, and the adoption
of sample plots would be justifiable.

5. Selection and Demarcation of Sample Plots.

It is hardly necessary to say that the sample plots should be as
nearly as practicable a true average sample of the entire crop.
Hence, before selecting it, the surveyor should go over the whole
crop, so that its average character may become clearly impressed
on his mind.

The following rules may be laid down for observance : —

I. — No sample plot should ever be selected on the edge of
the crop, for a true average will seldom be found there.
II. — On slopes presenting a wide range of elevation, or in
crops offering a variety of aspects and soils, several
sample plots judiciously distributed, should be selected.
III. — The form of the sample plot should be a long rectangle.
IV. — The boundary of the sample plot should be clearly marked
by blazing the trees immediately outside, or by
splashing them with whitewash.
V. — The aggregate area of the sample plots should be from

3 to 5 per cent, at least of the total area of the crop.
VI. — In mature crops, no sample plot should be less than
1 acre in extent, and <he minimum area should, as
a rule, be 2 — 3 acrejs. In young uniform crops, con-
taining a large number of stems, £ or even ^ acre
may suffice.

VII. — In crops of large extent, several plots of 1—2 acres each
are preferable to a single large plot.

4. Enumeration Survey of the Crop.

Bi-fore arry attempt can be made to calculate the quantity of
material in a crop, we must find out the number and respective

17

dimensions of the trees it contains, in other word*, make an tun-
meration survey. In an enumeration survey of a mixed crop, the
number and dimensions of the trees belonging1 to each species or
group of species of similar habit should be ascertained and recorded
separately ; all important or extensively distributed species should
be registered separately, the rest being classed into groups, each
group comprising species that resemble each other ill height and
shape.

As regards the dimensions of the trees, their girth is always
measured, the height, if that also has to be recorded, being
estimated with the eye. Since in every enumeration survey an
enormous number of trees has to be measured, it is not practicable
to register the exact girth of each tree, but to group the trees into
girth-gradations. The range of girth included in each gradation
will vary with the size of the trees forming the crop and with the
degree of accuracy sought. For the most accurate valuation,
survey, the following ranges are narrow enough : —

( large trees . . 3 to 6 inches-.
For a crop of < small trees . , 2 „

(^ very small trees . 1 inch.

Supposing 1 inch has been fixed as the range, then every tree
above 5£ inches, but not more than 6$ inches in girth, will be
classed as being 6 inches in girth; every tree above 6£ inches,
but not more than 7 3 inches in girth, will be classed as being 7
inches in girth; and so on — that is to say, all fractions not
exceeding one-half will not be taken into account at all, and all
fractions exceeding one-half will be considered as 1. And so on
with any other girth range. Here, in India, a girth range of as
much as 18 inches, established by Sir Dietrich Brandis in Burma in
1859, has been made use of in most of our working-plans, and
has been found to give sufficiently accurate data for the classes
of forest we have to work, and for the rough methods of working
them we are obliged to adopt.

As in extensive surveys it is generally found aiore convenient to
use diameter callipers than to measure girths with a tape, the cal-
liper should be marked in gradations of T7T of an inch, or a foot, or
other selected unit, so that the girth corresponding to measured
diameter may be read on the calliper; the following precautions
should however be observed. In crops consisting of fairly regular-

B

18

»

gluiped and not very large trees the measurement of a single
diameter may suffice, especially if, as tree after tree is gauged,
the diameters of two successive trees are measured in different
directions, more or less at right-angles to one another. Although
a matter of petty detail, it is necessary to say that the callipers
should be applied to the trunks of the trees properly, and the
diameter or girth read off before they are removed. The diameters
should be all measured at breast- height, and on hillsides this
height should be taken on the upper side of the slope. Breast-
beigbi has been assumed to be 4i feet ; but as the boles of
trees do not taper either regularly or very rapidly, it is not
necessary that this height should be exactly measured on the tree
before the calliper is applied. Sufficient accuracy is attained if the
measurer is careful to hold the calliper at the height of his chest
and the diameter is measured at any height between 44 and 5 feet.
When a tree divides into two or three main stems near the point
at which the calliper should be applied, each stem should be mea-
sured separately.

The enumeration survey should be effected over successive nar-
row strips, each strip being gone over once and in a direction:
opposite to that in which the immediately preceding strip has
been surveyed. On steep slopes it is convenient to ran the strips
horizontally and to begin at the bottom of the slope. The mea-
surers, furnished with callipers, gauge the diameter of the trees
and call out the figures read, which are at once noted in a pro-
perly-ruled field-book by the recorder, who is. usually himself the
surveyor in charge of the party. The number of measurers that
can keep one recorder fully employed depends on the density of
the forest, on the nature of the ground, on whether he is also in
charge of the party, and on whether.all or only certain classes of
the trees composing the crop are to be measured. The number of
measurers may thus, according to circumstances, range from 2 to
6, and even 7 and 8.

As the survey progresses' the trees measured are immediately
marked with a clearly visible blaze, which should not, however, be
deep enough to expose the wood. In order to make the blaze, each
measurer should he provided with a light short-handled axe. The
blaze should be made on the side opposite the area still remaining-
to be surveyed, so that when the next strip is being surveyed the

19

men can at once recognise up to what point the strip just com-
pleted extends.

The duty of the surveyor in charge of the party is to see that
the callipers are properly applied, the diameters or girths read
before the callipers are removed, and the blaze -made on the correct
side. When a division into height-classes is necessary, the surveyor
has also to judge with his eye the height-class of each tree as it
is gauged. Hence the advisability of investing one and the same
person with the duties of both surveyor and recorder.

The following is a sample of a convenient form of field-book—

RANGE COMPARTMENT

BLOCK ...

"S

c.Sj

Ci

5

Species
(or Hei

Number of trees.

a

5£

^§

"5 £
2

1

a

|«J

.H —

-1
fl

a

Species
(or Hei

Number of trees.

=

H

ght-class).

ght-cUss).

30

mi mi mi mi in
mi mi mi mi

43

16797

33

mi mi
mi mi

mi mi mi in
mi mi:

48

22-687

36

mi mi
mi mi

mi i

•

26

14-625

39

mi mi
mi mi

mi mi mi mi
mi mi mi mi

it

62

40-929

42

mi mi
mi mi

mi mi
mi mi

mi mi
mi i

56

42-875

45

mi mi mi mi
mi mi \ nu

34

29883

and so on

48

•

As the girth of each tree is called, a stroke is made in one of the
compartments opposite the figure expressing that girth. The first
four strokes are drawn upright, the fifth one obliquely across
<hem. Each group of strokes thus represents five trees of the

BS

20

girth against which it is drawn. Each full compartment represents
2U trees, and the whole line of five compartments 100 trees of one
and the same girth-class. After the survey is over, the total basal
area of the trees of each girth-class is calculated and entered in
the proper column. Lastly, the numbers of the tvecs and the basal
areas are totalled up.

Perhaps a more convenient method of recording the number of trees is the
one universally employed in France. It is as follows : — Each group, represent-
ing 10 tree*, consists of two upright rows of four dots each, joined by two
diagonal lines, which represent respectively the ninth and tenth trees, thus —

K=10; :\?=»; H=8| |: = 7; |;=6;

and so on

The subjoined form of field-book has been in general use in the North- West
Provinces and Oudh for more than ten years, and can hardly be improved upon
for enumeration surveys in which the classes include a large range of diameter
and the forest is very irregular. Some of its advantages are (i) that it requires
very little ruling, (ii) that it may be easily prepared from day to day by the
recorder himself, and (iii) that, as the width of its different columns and com-
partments can, for that reason, be varied to suit the composition of the crop to
be surveyed, a whole day's work, comprising several thousand trees, can be got
into a single opening of the book. The total numbers of trees of each species
(or group of species) and diameter-class is written in the right hand lower corner
cf its own compartment.

1- , . 1. 1 • •

-CoMPARTMBirr-

Specie*.

Ifuttber oftrett qf diameter exceeding—

Total

number
of trees.

6 inches,

«
12 inches,

18 inches,

24 inches.

Lut not exceeding—

841,

12 inches.

18 inches.

U inches.

K*QM1*H

and 10 on
1,«89

-

8»lo,

Chlr.

V

MUrrllmntoa*,

21

5. Palliation Survey ly means of the Average Tree.
The average tree, according to our assumption, is thnt tree
whose height and form-factor are the same as the average height
and form-factor of the crop, considered as a whple.

In order to obtain the average tree we must first of all deter-
mine what its girth is. This we shall know, if we know ite
hasal area. If c, a', h, and /, represent respectively the contents,
basal area, height, and form-factor of the average tree, then—

c — a /if;

But by formula (i) and (ii)

C^ ±HF
' » ™ iT"'

Therefore a k f = ~?l
and a = ^ - (since by hypothesis h f — IIP).

That is to say, the average tree is that tree whose basal area is
the average basal area of all the trees of the crop. This being so,
we can find out the girth of the average tree as soon as we
know the number of stems composing the crop and their total basal
area. Having obtained this girth, we fell and cube several trees
of that girth, and the mean contents of these trees will be the
contents of the average tree.

To obtain the above result, we have to assume that h f = HF.
This assumption is, however, true only when (1) the heights and
form-factors of all the trees in the crop, or, at least, the products
of their heights and form- factors are equalj (2) these heights and
form-factors, or, at least, their products, are proportional to the
corresponding basal areas, and (3) the mean of all these heights
and form-factors may.be taken respectively as the height and form-
factor of the crop as a whole. These three conditions can obtain
only in very regular crops, and even in such crops as many as three
or four of the trees of the average girth must be felled and

~ ~

measured in order to obtain a sufficiently close average for the
contents of the average tree. •

Resume. — When this method of valuation survey can be adopted
the following procedure should be followed : — Measure and register
the trees in girth-gradations embracing a range of 1 inch. Then,
with the aid of tables, calculate the aggregate basal area of the
trees in each class, and total the whole. Divide this total by the
number of trees, and the quotient will be the ba-cal area of the

22

average tree. From this deduce the girth of such tree. Lastly,
fell several type trees of this girth, and find out their contents
accurately. The mean of their aggregate contents will be the
contents of the average tree, and this, multiplied by the number of
trees in the whole crop, or, which comes to the same thing, by the
ratio of the total basal areas to the basal area of the average tree,
will give the contents of the crop.

6. Survey by Girth or Heighl-and Girth-Classes.

In a forest crop there is never any constant relation between
the heights, girths, and form-factors of the component trees. Of
these three elements, two of them remaining the same, the widest
fluctuations are presented by the girth, the other two varying
within much narrower limits. In regular crops the average height
of the stems comprised in each class varies directly as the girth
of that class, the thicker stems being also the taller and vice versa ;
BO that we may consider the height as a function of the girth.
If the crop is very regular, the differences of height between the
component trees will be so small that the height of the crop may
be expressed by a single figure. In irregular forests, on the other
Land, there can never be any constant relation between height and
girth, for the thickest stem may be the shortest of all.

As regards form-factor, it may be regarded as a function of
height and girth combined, for its variation depends on the same
circumstances which govern the variations of height and girth and
especially on the ratio of height to girth. Hence the form-factor
may be assumed as being nearly equal for trees of like girth and
height. But even in regular crops measurements of sample stems
of one and the same girths may give differences of from 4 to 6 and
even 10 per cent, in the form-factors'-

We will now investigate the different conditions that may
present themselves in the survey of crops or of a group of trees
composing a girth-class, which may be regarded for the present
purpose as a small crop by itself.

Starting with the element of girth, we have two broad cases :
The girths of the trees may either (A) be different, or (B) be
one and the same.

A. GIRTHS DIFFERENT. Under this head we may have four
separate cases as follows :-—

(a) Height and form-factor constant, «.*., form-factor a func-

23

tion of keiykt. We know that a = A (page 21), and

»
£=<? ^/ (formula iv, page 12). In other word*

a

the sample tree of average basal section is a true aver-
age tree. This case obtains only exceptionally in *
whole'crop, but is always approximately true for the
groups of trees composing its several girtli-chisses.

(b) Height and form-factor a function of the girth and

hence variable, or height constant, but form-factof
variable and a function of girth. — The former con-
tingency may, as a rule, be assumed as always prevailing
in regular crops of one and the same age and origin ;
the former can very seldom happen. In either con-
tingency only girth-classes should be adopted.

(c) Height irregularly variable, the form-factor being a func-

tion of height. — In this case height-classes must also
be formed.

(d) Height and furm- factor both irregularly rariable.—f\i\*

is generally the case in irregular crops composed of
trees of various ages. Here also girth and height-
classes must be formed, and in addition several sample
trees should be measured for each class.

B. GIRTH CONSTANT. This is always true of girth-classes,
and may be assumed to be true also of girth gradations.
"We have three distinct cases as follows : —

(a) Height, and form-factor constant or both of them functions

of girth.— In such a contingency every component
tree would be a correct average tree, but this is never
exactly the case even in the most regular crops.

(b) Height constant or a function of girth, form-factor vari-

able.— This is the case generally met with. In order
to obtain a correct average form-factor several sample
stems must be measured.

' (c) Height variable and form-factor a function of height.— In
this case height-classes must be formed, thus reducing
the conditions (for each height-class) to those of case
(a).

. 24

The preceding detailed study indicates the several cases in which
it is necessary to form only height-classes or ouly girth -classes,
or comhiued height and girth-classes. Every crop to be sur-
veyed must be divided up in this manner into a small number of
distinct groups, for each of which the corresponding sample tree,
possessing the average basal section, may be taken as a true repre-
sentative in respect of both height and form-factor.

In the constitution of girth-classes one of three systems may be
adopted, riz. —

I. — THAT OF EQUAL GIRTH-GRADATIONS. — The number of grada-
tions are usually from three to five, and the figures of the
enumeration survey at once indicate the number of trees
to be included in each gradation. In this system the
disparity between the numbers of trees in the several
classes is always very great.

II. — THAT OF NATURAL SIZE-CLASSES. — In the most canopied
crops at least three classes of individuals are easily recog-
nisable— (1), a class of small stems with restricted crowns
ana below the average height of the crop; (2), a middle
class with freely developed crowns and of the same height
as the crop ; and (3), a class of thicker stems with crowns
rising above the general mass of the crop. To these classes
may be added two others, viz., that of suppressed indi-
viduals and that of old standards or trees of much o-reater
age than the rest of the crop, arvd conspicuous by their re-
latively great size. These several classes are generally at
once recognisable from the figures of the enumeration
survey.

Ill,— THAT OF EQUAL GROUPS.— If ,natifral size-classes are not
distinguishable by their relative thickness of stem, then
it ie advisable to form classes containing approximately
the same number of stems. In 4his method the range of
girth included will obviously differ from class to class,
and the method is therefore to this extent the very op-
posite of I.

The numher of sample trees to measure in each girth-class
will depend (i), on the relative proportion to the entire crop which
the claes represents either in respect of number of stems or of cubi-
cul contents; (ii), on the uniformity of the component trees in

25

respect of height and shape, i.e., the case of greater or lesg diffi-
culty in selecting correct representatives of the class in question,
and (iii), on the possibility of errors due to an insufficient number
selected in one or more classes being compensated by excess or
deficiency of results obtained for the other classes. Obviously the
possibility of such compensation exists only if each sample stem
represents approximately an equal fraction of the total contents of
the crop.

As a rule, the number of sample trees measured in any girth-
class is determined by the proportion either (a) of the number
of stems, or (b) of the aggregate basal sectional areas comprised in
that class. The adoption of the latter of these t\vo bases of calcu-
lation is always more likely to give correct results.

The grouping of the trees into height-classes must take place
at the time of the enumeration survey, and be a part of that
operation ; their separation into girth-classes may be effected after
the survey with the help of the figures which it furnishes.

To illustrate what has been said in this section we append
below a specimen of a survey by girth-classes.

"Forest Kanjatra Sample Plot (a) 264' x 165'. Age 135 years.
Compartment 97. Area 1 acre. Density of stock 0'9.

n

.

SAMPLE STEMS.

S*?

s

<M

•glil

M

Basal are.i,

0

^

Girth in ir
breastrhe

Species, Spruce.

Number of

in square
feet.

1-

GJ

rt
2.

"3
3

m

-<-> .

G~Z

s ?

•4* '

K

^

^

|

B

Remark*.

30

nu

In.

Sq. ft.

C. ft

Ft.

Vrn.

1

/

6

2314

\

33

Hil

8

3-781

III

36

HU
Hil

1

I.

11

6-187

39

•6601

27-33

92

•46

135

39

HU
HU

Illl

14

9-242

brar

chea

•957

=8J

X

42

HU
HU

HU
II

17

13016

•

26

.

SAMPLE STEMS.

i

• <*-«

<C .
0 -u

1l

- ®

Species, Spruce.

5

Basal
irea, in ,
square

1

u

S

1

g

ifl

o

feet.

i Va r*

^

.

S

"** K

»2

^3

'A So

pr*

S

g

fj

S

-tJ
f-.

o

55 cs c

"S
B

1 §

3 M

w /

«/

w

i

16

14*062

1 1

48

-

w /
/w/

w
///

19

19'0<X

1 51

1-1289

35-88

111

•43 1'

nu nu mi

51 I uij M//

/7W //"

24

27'09

j 54

1-2656

59-00

111

•42 1

361

M

/«/
/«/

•

IHI
1

•

16

20-25

2-3945"

L12-88

57

w
nu

f

16

22-56

bran

ehes '

1-52=

«.

60

nu
nu

""

i 14

21-875I

63

nu n

17

29-28fcl \ (.

>9 2-0664

95-05

115

•40

134J

66

nu
nu

10

18-9

361 '

r2 2-2500

101-79

116

135J

69

nu

'

11

22-7301

\

72

nu

in.

7

1

15-750

4-3164

196-8

u

9

75

nu

7

^17-090

branches

8-46-4-31

Vo

••M

n

— i •

_

-

78

in

3

7-922

81

HI

3

8-543

Girth Class I.
Ill

• »t f. HA-

5
10
5

6 34-570
5 124-843
8 120-226

Total . 21

9 279-630

27

Calculation of Quantity of Wood.
CL= 27-33 x -M^2= 1)431 c. ft
C2=112«88x l^j.= 5,885 „
C3=196-84x -^gj = 5,482 „

12,798 „ = wood of stems.
Add 4% 552 „ = „ „ branches.

Total Wood . 13,350 „

Calculation of Average Stems.
— 5g— = G'6171 sq. ft. and gi=38 in.

124-843 - .10 rv A

a2= -105- =1'1890 „

120-226

w gs=

7. Survey by Girth-gradations. Methods of Draudt, Uriel, and
Ear tig.

\Vhenever it is impossible to determine accurately the average
tree even for separate girth-classes, it is necessary to adopt as
the basis of survey girth-gradations (see page 13, I A ay),
which obviously require the measurement of a larger number of
sample stems than the method just described, the number for each
gradation being proportional to the total number of stems com-
prised in that gradation.

Several methods of survey by girth-gradations have been
devised, but we will descri be here only those of Draudt, I 'rich, and
Robert Hartig : —

DRAUDT'S METHOD.
In this method a certain proportion, say the Mh part, of all the

28

frees in each gradation, and, therefore, also in the whole crop, are
measured as sample trees, the number of such sample trees being
therefore

= — = say 5.

z

Now as the sample trees represent the 2-th portion of the whole
crop, not only in number but also in respect of contents and basal
area, we have

A => a z and C = c z = c - = c — .

* a .

As the 2-th part of the number of trees composing1 a girth
gradation may not be a whole number, and we cannot measure a
fraction of a tree, it is best, in calculating the cubical contents of
all the trees in a gradation, to use the last of these equalities, which
enables us to measure up whole stems only, and also renders it
unnecessary for the sample stems to be exact average stems for the
gradation in question.

In practice the procedure is as follows : —

An enumeration survey is effected in classes having a sufficiently
wide range of girth (say 6 inches). v This being done, the
figure z is determined, and the number of sample stems to be mea-
sured for each gradation is then the nearest integer in the expression
* M

— . But if — is a very small fraction, as many gradations are

t z

lumped up together as will give an e sample tree ; and when this
if done, the basal area of the sample tree is determined in the same
way as when several girth -classes of' the enumeration survey
are lumped up together to form a new girth-gradation (fee the
figures at the bottom of the example at end of the preceding
section). The girths and basal areas of all the sample stems
are then carefully registered, vand their'cubical contents accurately
mounted nnd expressed, either in otoe lump figure or in separate
figures giving the respective quantities of timber, fire-wood, etc.
Lastly, the contents of all the trees in each gradation are calculated

29

by the formula (?=*—, If the trees falling under one and the
same girth-gradation are of very different heights, this fact
must be borne in mind in selecting the sample trees, or the girth-
gradation may be divided into sub-classes according to height and
each sub-class treated in the way described.

The great advantage which Draudt's method offers is, that all
the sample stems for the whole crop may be measured up together
and their contents determined, not only in one lump figure, but also
according to the different classes of produce they yield, thereby
enabling us to estimate by means of a few easy multiplications the
contents of the entire crop.

URICH'S METHOD.

We have seen that in Draudt's method fractions in the quotient
of ( j \ are got rid of by taking the nearest whole number, but where
the quotient is much less than unity, several quotients are added
together, and the result worked out for a group of girth-gra-
dations. This procedure is obviously not quite logical, and hence
Urich has modified it so as to secure greater cousistency. He
adopts the latter principle throughout, and his system is according-
ly always to carry over all fractions to the next class. The sample
trees are hence seldom required to represent a separate girth-
gradation, but nearly always a group composed of the whole or
portions of two or more such gradations. The girth of the
sample trees for each group is accordingly determined by the pro-
portion of the respective numbers of the several gradations compos,
ing the group. When the formula C = c -- is employed, it is
not so necessary that the sample tree should be representative of
the group in respect of volume as in respect of height and form-
factor.

The advantages of Urich's method are the same as those of
Draudt's.

In its valuation surveys the Commission for Forest Research in
Germany adopts five girth groups, and fells from two to three
sample stems for each group. This is a combination of CJricVs
principle with the method of girth-classes.

30

HARTIO'S METHOD.

In this method tbe groups are so formed that the component trees
Aggregate equal basal areas.

The procedure is as follows : —

First decide what number of sample trees (£) or of groups of
trees (G) we require, and theu determine the aggregate basal area
to be included in each group (this area = TT or -77). Now form
the groups, beginning with the smallest class of trees. Next the
girths of the several sample trees are either fixed approximately
by inspection, or rigorously determined by means of the formula

a — ~- . Each sample stem is then measured by itself, and the
n

contents of the corresponding group ascertained with the help of
the formula C = c — . Or the contents of each sample tree may

be considered as the contents of an imaginary cylinder of the
same base as the tree, and the corresponding height of the cylinder
obtained from the formula li — —^ the total contents of the crop

OJ

being the product of the total basal area oE the crop multiplied by
the mean of all the cylindrical heights thus obtained.

In this method the larger stems obviously compose, number for
number, more groups than the smaller ones, and the sample trees,
although of course samples of the corresponding group, are not
samples of the crop considered as a whole. The contents or yield
in different classes of produce of the sample trees cannot hence be
worked up together in one place, and the main advantage afforded
by Draudt's method is thereby lost. Comparing the two methods,
Hartig's may be employed when as accurate as possible an estimate v
of the contents of the crop is required with the help of only a few
sample trees, whereas Draudt's should be adopted when it is possi-
ble to fell a larger number of sample trees and an estimate of the
yield in different classes of produce is required.

EXAMPLE OP THE THREE METHODS.

To render the preceding explanations clear, we proceed to show
below, by means of a comparative .parallel statement, how to use
the three methods of valuation just described. We take the case
of a 1'7-acre sample plot of beech.

3t

Distribution of the Sample RtMM

Result of
Enumeration survey

82

In the preceding1 example, in applying Brandt's method, the
tret-s respectively of the four highest girth-classes heing separately
very much less than Z (75), are lumped up together for the
purpose of the valuation.

To complete illustration of Urich's method, let us suppose that
we have selected and felled ten sample trees of the respective
average girth, and that they furnish the following figures : —

Total basal area of the 10 trees .

Yield in timber „ . .

„ „ fire-wood „ . .

„ ,, faggot-wood ,, . .

Then^__. 61 7-228 _

a ~ 11-62 — 53'1, and the contents of the whole crop

are-
Timber = 98-53 x 53 1 = 5,232 solid c. ft.
Fire-wood «= 856-40 x 53-1 = 45,475 stacked c. ft.
Faggot-wood = 84-7 x 53-1 = 4,498 „ „

The completion of the valuation by Hartig's method is effected
in the following tabular statement:—

11-62 sq. ft.
98-53 solid c. ft.
856-40 stacked c. ft.
84-70

THB SAMPLE STEMS.

Contents of each

tiro up.

Content i.

a

Eleight

of

number,

-

Basal
area.

wood.

Fag-
got
wood.

Total.

cylin-
der.

Large
wood.

Fag-
got
wood.

Total.

A

a

I

-

i
•

I

to

111.

Cubic feet solid.

ft.

Cubic fee? solid.

1

3'

•3906

17-2

84

20-6

52-7

2,734

540

3,274

62-092 -f- -3906 = 158-96

2

ft

•5625

27-5

65

330

' 58-7

3,025

605

3,630

61-875 -5- -5625 = 110-00

3

42

•7656

40-0

8-0

48-0

62-7

3,226

- 645

'3,871

61-749-*- -7656= 80-65

-}

it

•7656

40-0

80

48-0

627

3,240

'648

3,888

62-016 -r- '7656= 81-00

f.

U

•9184

53-2

8-4

616

671

,3,569

563

4,132

6 1-609 -f- -9184= 67-08

f

u

1-0000

54-7

8-5

632

63-2

3,391

627

3,918

62 000 -i- 1-0000= 62-00

7

."ii

1-128H

628

8-8

716

63-4

3,406

±77

3,883

61-234-s-l 1289 = 54'24

s

:, i

1-2656

71-0

9-4

80-4

63-5

3,449

457

3,906

61-484-5- 1-2656- 4858

9

Si

1-5625

880

10-0

98-0

62-7

3,473

395

3,868

6 1-671 + 1-5625= 39'47

K

7'

2-1267

145-0

144

159-4

74-9

4,288

426

4,714

62-S90 -i-2-1267 = 29'57

Total

x
631-6

33,801

5,283

39,084

Or we may use the formula C = A II F ; whence total contents
= 617-223 x '= 38,984, cubic feet.

33

8. Number, selection and Measurement of the sample trtt».

The number of sample trees to be measured will depend on the
method of survey adopted, and on the size and the decree of uni-
formity of the crop. The method of survey- which requires the
smallest number is that of the average tree, 2 or 3 stems sufficing
if the crop is regular. In survey by girth-and-height classes from
6 to 15 trees must be measured. Draudt's method requires from
10 to 15.

Since special circumstances, independent of the method of sur-
vey, often limit the number of trees that may be felled, the choice
of the method of survey must, in such cases, be regulated by the
number of trees which it is permissible to fell. It must, however,
be borne in mind that the greatest accuracy in measuring the
sample trees can never compensate for the smallness of their
number. Averages derived from the measurement of a large num-
ber of trees will, even if the measurements have been made less care-
fully, be more trustworthy than those obtained with the help of a
smaller number of trees. There is less error in etlimating the
contents of a large number of standing trees with the aid of Volume
Tables, or by means of the nc^-height, than in measuring up an in-
sufficient number of felled trees in the most accurate manner potable.

The care exercised in selecting the sample stems must be in
proportion to the smallness of their number. They must be as
nearly as possible correct representatives of their class in respect
of height, form and branching, and all forked and otherwise
abnormal trees should be avoided. Moreover, the boles should be
perfectly regular at the places where the girth has to be measured.
" "When a diameter calliper is used, at least two diameters should be
taken at each place to obtain a correct average. When the number
of sample trees to be felled is large, they should be situated at
different points of the entire crop. The sample trees should always
be chosen immediately after the enumerati&u survey has been com-
pleted, and by the same person who has conducted thia survey, and
who has therefore a correct and vivid general impression of the
character of the crop and the component trees.

The cubing of the sample trees should be effected as accurately
as possible by the rules already laid down for the measurement of

felled trees.

c

34

9. Valuation survey by means of the v\c\\t-Iteight.

The lower girths of all the trees of the crop being known
from the enumeration survey, the total basal area A is calculated
therefrom. Next the rzc^-height of a sufficient number of sample
stems should be observed, and the average of all these figures
assumed as the rtcM-height Ii, of the whole crop. The contents of
the crop are then obtained from the formula

C=l AHr + Afi0.

The contents of the branches must be added as a percentage,
which must be taken from tables, or deduced from the results of
previous surveys of similar crops.

The crop may of course be divided into girth-and-height
classes, and the average ric4l-height determined for each class
separately.

10. With the aid of tables of form-factors or volumes.

The girth of all the trees of the crop are already known from
the enumeration survey. The heights of a sufficiently large num-
ber of trees must be measured, and the average heights which
correspond to different girths determined therefrom. The fol-
lowing is an example, exhibiting a convenient mode of arranging
and manipulating the various figures : —

Girth in inches

24

30 36

4?

48 54

60

66 72

78

Height in feet.

82

98

104

-Ill

111

115

114

115

120

J17

Species— Beech . .

...

96

• 1C7

114

112

112

115

112

117

...

Total height

...

194

102

111

...

114

...

116

117

82

313

336

223

341

229

227

353

Average height .
Corrected height .

82
82

X97

97

104
104

112
110

112
113

114
114

114
114

114
114

118
116

117
117

35

The calculation is completed thus (in the field-book itself of the
enumeration survey) —

Comportment 11.

1

CoHtyxTi.

o

q-

g

•g

ja

Species— Beech.

o

S
jj

Per tr«».

Total.

Reuurk*.

5

1

tC

1

5

S

E

Build cubic fe«t.

24

-Si

50

82

13-9

695

30

T2 i

114

97

258

2,937

36

§ ® .

158

104

41-2

6,510

42

£-*->£>

156

110

60-0

9,360

48

-2*5 a

128

113

82-1

10,509

54

ZZ* 00 00

66

114

I 1070

7,062

60

^ 0 0

42

J14

133-0

5.586

66

i^'is

19

114

163-0

3,097

72

W s g

12

116

198-0

2,376

78

3 a

5

117

237-0

1,185

•.

750

...

...

49,317

The " contents per tree " are obtained from the tables of form-
factors or volumes, as the case may be.

11. Valuation survey by ocular estimate.
Two modes of procedure may be adopted —

1. — The contents of each tree of the crop may be estimated, the
contents of the entire crop being then obtained by a
mere sum of simple addition. Such a procedure, how-
ever, takes as much time as a complete enumeration
survey based on the actual measurement of the girths
of the trees, and once the girths have been measured
and recorded, there is very little extra trouble in measur-
ing the heights of a few trees and obtaining the contents
of the crop by the much ncore trustworthy method of
valuation by the richt- height, or with the aid of tables
of form-factors or of volumes.

1L The contents of the whole crop may be estimated en bloc.

An intelligent forester who has had long local experience
in the clean-felling of coupes can often give a fairly
accurate estimate of the quantity of produce standing
per acre, but the best man is nevertheless liable to make
an error of as much as 50 per cent.

36

12. Valuation survey by means of tables of yield.
In this method the surveyor must determine three essential
points : —

(a) The quality of the soil and locality.
(6) The density of the crop.

(c) The age of the crop.

The older tables of yield drawn up in Germany omitted the first
point. In the tables recently issued, for each forest or class of
forest a certain convenient number of classes of soil and locality are
established according to the height attained by the trees in each.
Hence, for any particular crop in question, we have only to ascer-
tain the average height of the trees, iu order to know at once the
quality of the soil and locality.

The age of the crop will be known from its past history, if it
has one going back far enough ; or it must be ascertained by ring-
countings, if some constant relation exists between the number of
concentric rings and the number of years in which this number of
rings is produced. Otherwise there is no means of obtaining it
with any degree of accuracy.

The density of a crop is an extremely difficult thing to estimate
with sufficient accuracy. It can, of course, be determined by mea-
suring the girths of all the trees and thus obtaining their
aggregate basal area, but this means almost as much work as far
more trustworthy methods of valuation survey with the aid of
sample trees.

13, Choice of the method of valuation survey.

Prom what has already been said in describing the varfons
methods of valuation survey, it Is clear that this choice in any
given case depends—

(a\ on the required degree of accuracy,

(b) on the nature of ^th'e crop, and

(c) on the number of individuals that may be felled as sample

trees.

If the money-value of xa crop is sought, as would be required if
the crop is to be sold or the forest expropriated, or if the owner
wished to obtain accurate statistical data regarding his property,
Urich'g method should be employed. For the purposes of a work-

37

HILT plan a less exact method is admissible, particularly as regards
the younger crops, which must he surveyed over again when the
plan is revised. Hence for such crops a complete survey is seldom
necessary, and in the sample plots the method of rtoi {.height and
those based on tables of form-factors or of volumes or of yield may
be adopted. But where a certain degree of accuracy is required,
the method of survey by the average tree may be employed in
regular crops, a higher degree of accuracy being secured by the
establishment of girth-classes and the highest by U rich's
method, which is moreover the only one to adopt when it is re-
quired to estimate the yield in the various marketable classes of
converted wood.

The establishment of height-classes gives a great deal of trouble
and extra woik. It should be avoided whenever possible, that is
to say, as often as the heights of the trees composing the crop do
not exhibit any marked irregularity.

Chapter V.

On the determination of the ages of trees and crops.

The determination of the age of trees and of crops is a problem
which often presents insurmountable difficulties to the Indian
forester, since not only are the ring markings indistinct and
sometimes indistinguishable, but so many of our species form more
than one concentric ring of wood each year, and there is nothing
to prove that in their case the number, of rings is one and the same
for each year. The following remarks hence apply only to species
which invariably form a single distinct concentric ring of wood
each year.

1. Determination of the age of standing Ircet.

The ages of individuals of most of our conifers can, as long aa
they are branched down to the ground, be accurately determined
by counting the number of annual shoots. The age of other treei
can generally be told to within 10-20 years by a forester pcssessed
of large local experience. But the most certain way of ascertain-
ing the exact age of a tree is to use Pressler's borer, which should
be long enough to reach, or all but reach, th« centre of the trunk.

38

This instrument (see figure) is a gimlet, consisting of a tube (G)
with a very sharp-cutting edge (E) .

To render the instrument easily portable, the gimlet portion G
can be taken off and put into the cylinder CC, which is hollow,
and the caps of which unscrew off.

As the tube is forced into the trunk of a tree, a cylinder of
wood is cut out by the tube. On withdrawing the gimlet, the
cylinder of wood can be easily pushed out of the tube and the ring-
markings on it counted. When the borer does not quite reach
the centre of the tree, the age of the remaining portion of the trunk
can be estimated with sufficient accuracy.

If the conditions of the forest have not materially altered since
the appearance of the trees experimented upon, the ring-countings
will give also the age of individuals of "any girth-class smaller
than the class to which those trpes belong.

2. Determination of the age of felled trees. ,

It is scarcely necessary to say that, under the assumption made
at the beginning of this chapter, the required age is accurately
determined by counting the number of concentric rings on the
section of the stool or trunk.- The remark made in the last para-
graph of the preceding article holds good here also.

5. Determination of the age of entire crops.

If the crop is regular, its. age is practically the age of the mid-
dle class of stems composing it, and. it will hence suffice to deter-
mine the age of one of those stems, or, to be on the safe side, of a
few of them.

39

If the crop is irregular, the problem becomes more or less com-
plicated. Several stems of the different girth-classes present
must be examined, and when the respective ages of the several
classes have been determined, the question to be solved is how to
obtain the mean age of the crop from them. To take the mere
arithmetical mean of the several ages without reference to the
respective areas occupied by them, or to the quantity of material
each represents, would evidently be wrong. We now proceed to
investigate different methods for obtaining the true mean age
which may be defined as that age at which a crop of uniform age
would, under the same conditions of soil, locality and species, have
produced the same volume of material as the actual crop contains.

Let vv v2t t'3 = respectively the volumes of the several

classes aged, respectively, yv y2, y$ years, and Y = the re-
quired mean age = the age of the imaginary equivalent crop. By
hypothesis the mean annual increment of both crops is one and the
same ; let this increment = 7. Then —

1 V i

II = V} + V2 + t>3

and 1 = — + .- ' + — ;

yi yt ys

Hence f- + - + -8- I Y = 0, + v9 -f v. .

\y\ ya y» '

and r = Vl + r* + *> Forraula(v).

®1 i *"3 i *°J

yT ~ ys y3

Expressed in words, the preceding formula would run thus: —
To obtain the mean age of a crop composed of trees of diverse ages,
divide the total volume of material on the ground by the sum of the
mean annual increments of the several age-classes.

Since the age of trees is, at least approximately, a function of
their girths, the girth-classes may be considered as coincident with
age-classes, and the words "girth-classes" may be substituted
for <l age-classes " in the above rule.

Let us now investigate another formula' for the case in which

the respective areas otlt a2, «3 occupied by the diameter-classe«

are known. If iv i2, ?3 - = respectively the meun annual

increment per unit of surface in the different areas «,, «s, as ,

then we have

"l *l '1 y\ > V2 ~ 22 .'2 ' 3 ""

40

Substituting these values of v and - in formula (v), we have

*1 ;l «S ** «3 '3 ......

If in the above formula, tj = »2 = ?3 = ...... , we have

Y = «i*i + *,*, + •» 9, ...... Formula

The employment of this formula presupposes a knowledge of the
respective areas occupied by the different age-classes. It is, there-
fore, not adapted for the calculation of the mean age of a crop
composed of trees of various intermixed ages, but its special use is
for the determination of the mean age of several crops considered
together, or even of an entire working circle, and it is generally
employed for this purpose. It gives the same result as Formula
(v) when Y is approximately the age at which the highest mean
annual increment occurs.

Il now remains to investigate a formula for finding out the
mean age of the crop when we know only the numbers nv n2, »3 ......

of trees included respectively in the several classes whose ages are
7i> ?2> Js ...... ^y *ne ordinary rule of arithmetic for obtaining

averages, we have the mean age of the crop

.. . Formu|a vii.

If this formula is to yield the same result as Formula (v), it is
necessary that the mean annual increment of all the diameter-
classes should be one and tha same. For iefc £t, £j, £g ...... = the

mean annual increments of,the average trqes of tbe several cla«s.is v
then

and by formula (v)

Xi ni + X* »* + X* «s

Hence, in order that formula (vii) should at the same time be
true, we must have

X\ = X*. = Xs ......

For formula (vii) to give the same result as formula (vi), it

41

is necessary that the number of trees per unit of area should be one
and the same for all the age-classes. For let PP r>it u v ...... be res-

pectively these numbers for the several classes, then

and by formula (vii)

' ai v\ v\

• ai "i "*" aa va + as "3 ......

by formula (vi)

T also = *'•*« + "2^2 + g«*s •••••• .

«1 + «« + «3

and we must hence have

"i = V2 = vs = .........

It has now been shown that formula (vii) holds good only on
condition (a) that the mean annual increment is one and tl>e same
for all the girth classes, and (b) that the number of trees per
acre is one and the same at all ages, assumptions that are incom-
patible with actual facts. Hence the employment of this formula
should be avoided.

If in formula (viij we assume that «L = n2 = n3 ...... = n ;

that is to say, that the number of stems in each age-class is the
same, we have

M

=. the arithmetical mean of the ages of the sample trees. Now, in
Urich's method of valuation survey each sample tree corresponds
to one and the same number of trees in the crop. Hence if, in
working according to that method, we deduce the mean age of the
crop by taking the menn of the ages of the sample trees, we obtain
the same result as if we had adopted formula (vii}, which has just
been shown to be incorrect.

Nevertheless, as by far the easiest way of determining the mean
age of a crop is to take the simple arithmetical mean or' the ages of
the sample trees, let us examine under what conditions such a
procedure wonld give correct results.

Let us then suppose th'jit in this special casj? the mean nge of
the crop is equal to the arithmetical mean of the ages of the sam-
ple trees. Hence

+ .y + V ..-.» - ' ri + «>?+'•« + •'•••• TK For.

+ ^ 4 _!> + ^

y* r* ' /*

42

Multiply both the numerator and denominator of the first side of
the equation by such a number z that the numerators of both sides
may be equal. Then the denominators will also be equal.

Hence 2 yl + zy^ + z y$ zyn = *>i + ** +vs ... + vn,

p, fg I ^a PU

and z + z + z up to n terms = - . H H — + —

These two equalities are possible only on the condition, that the
first, second, third ... terms of one side are equal respectively to
the first, second, third ... terms of the other side, that is to say,

But volume (0) = basal area x height x form-factor = alif,

. #j«l/l ^2"o/2 ^8 3 ' 3

Now we may assume that in one and the same crop the mean
annual increment of height x form-factor ( viz., J- \ is approx-
imately equal, that is that

y\

Hence <^

These conditions aue fulfilled by the distribution of the sample
trees in Hartig's method of valuation, so that in that method the
mean of the ages of the sample treed gives the mean age of the
crop.

For the employment of formula (v) a survey by girth-classes
is necessary in order to be able to determine the volume of material)
contained in the several classes. Tb;s is' not possible by Urich's
method.

It may sometimes occur, when only a few sample trees are
felled, that the smaller are found to be'older than others of larger
girth. In such a case we nlay still take the arithmetical mean
of the ages on the not improbable assumption that there is a wide
difference of ageg between the several individuals of each of the
girth- classes, and that for bue sample tree that gives too high
a figure for the mean age of its class, there is another which gives
the sama amount of compensating error on the other side — errors
of excess and defect thus cancelling each other. The smaller the

43

difference between the ages of the oldest and youngest trees in one
and the same girth-class is, the more approximate to the true
mean age of the crop will be the arithmetical mean of the ages of
the sample trees. And the result will be all the nearer, when the
number of sample trees taken is, is large and the more closely the
system of forming the stem-classes approaches that of Hartig's
method.

Chapter VI.

Determination of the rate of increase of individual trees and
of canopied masses of trees.

Single trees increase in height, girth, basal area and volume.
Similarly, crops of trees increase in height, aggregate basal area
and volume.

The current annual increment of a tree or crop is the amount by
which it has increased during the past year.

Periodic increment is the amount of increase gained during a
period of several years.

The total increment of a tree or crop is its actual volume at any
given time (the sum of all the annual increments up to the given

time).

The expression mean increment, when used without any other
qualifying word, may refer to merely a period, or to the entire
age of the tree or crop, or to the age when the tree or crop becomes
exploitable.

In determining the increment for a single year, it is customary
to assume for it the figure of. the .mean increment for a short
period of years, because the increment for a single year is not
only composed of factors too small to be accurately measured, but
is subject to disproportionably large fluctuations from year to

year.

For many purposes the increment is conveniently expressed as a

percentage.

2. Increment of indiv iduai.tr ees.

A. Bate of-increate in height.

The rate at which the height has increased can be determined
at once in the case of most conifers, since the length of each annual
shoot is apparent owing to their peculiar mode of branching.

In the case of broad-leaved trees, the stem must be cut acroat at

44

the top. If the section shows n annual rings, the portion cut off
is « years old, and the point must be found at which there are
exactly » such rings, the number immediately below being n + 1.

"When the increment for every stage of the existence of the tree
is sought, the stem must be divided into equal sections, say of 5
feet length. The number of annual rings counted on the upper
surfikce of each section will give the number of years' growth
above it, and the difference between the age of the whole tree and
this number will give the number of years which the tree has
taken to attain the height at which the rings have been counted.
The preceding mode of procedure must be adopted even for conifers
when the lower branches Inure fallen off.

It is evident that if the number of concentric rings does not
correspond in any way with the age of the tree, the determination
of the height increment, except for recent years the shoots corre-
sponding to which are apparent at the top of the stem, is impos-
sible.

B. Rate of increase in diameter and basal area.

As it is impossible, or at least very difficult, to ascertain the
girth-increment by direct measurement, it must be deduced from
the diameter increment by multiplication by tr = 3'14. To
obtain the diameter increment, measurements are generally made
at the usual height of 4J foot, or breast-height, but they may be
made instead at the middle of the tree, or, if great accuracy is
required, at several places.

Jn the case of trees forming one or a regular number of rings
each year, the diameter increment of a given year or. period of
years is found by measuring the thickness of the layer of wood
put on during that interval. The thickness of this layer must be
measured on several pair of radii,, each pair belonging to one and
the same diameter. Twice the mean of the several measurements
will give the increment sought. A single measurement is
justifiable only when both the shape of the stem and crown and the
branching are extremely regular. If the tree cannot be felled,
I'ressler's borer should be used. The increment of the sectional
area at any height will be given by the formula

? (/;»-</») = 0-7854 (D + d) (D-4)

45

where d = diameter at the beginning of the year or period, and
D = diameter at the end of the same.

As regards trees, the different concentric woody layers compos-
ing which are not distinguishable, or, even if distinguishable, are
not the same in number for each year, the procedure just described
is not applicable. In their case the only plan to follow is to measure
trees of known ages and growing under identical conditions of
soil, locality and leaf-canopy, and deduce the required increment or
increments by comparing the various figures thus obtained. \Vhere
trees of different known ages are not obtainable, there is no alter-
native but to select one or more plots representing the average
characters of the entire forest in respect of soil, locality, composi-
tion and density, and containing between them individuals of all
the age-classes. These sample plots should be properly thinned
from time to time, so that none but trees growing without any un-
necessary check may come under observation. In order to secure
this object more effectually, no suppressed or overtopped trees should
be measured, nor even those that have their tops exposed to the
sky if their crowns have a restricted development owing to lateral
pressure from their neighbours. Subject to these exceptions, the
girths of all the trees should be taken at regular intervals of one
to five years, according to the rapidity of growth of the trees.
Unless the trees increase in girth very rapidly, the interval should
not be less than two years, for not only is the increment of a single
year a small quantity difficult to appreciate, but various disturbing
causes, of which the splitting of the bark and its falling off in
scales and the varying amount of moisture in it, are the chief,
combine to mask or exaggerate it, as the case may be. The mea-
surements should betaken each time along the same circumference;
and in order to secure this each stem should be encircled with a
steady line of white paint about half an inch wide. The most last-
ing paint is zinc white. According to the nature of the bark, the
ring of paint must be renewed at longer or shorter periods. The
entire ring should lie in the plane at right angles to the axis of the
bole at the height at which it is painted. The rings should be
painted at 4i feet from the ground, but if there is some marked
irregularity of growth at this height, two rings may be painted
respectively above and below this height at an equal distance from
it in accordance with the principle laid down in rule (/) on page

46

4. In measuring'the girth, the tape should be laid along either
the lower or the upper edge of each ring, never along both indif-
ferently. Experience has shown that it is most convenient to
measure along the lower edge. The tree should be numbered con-
secutively for recognition, and the numbers should be painted on
the bark close to the ring. Labels, of whatsoever pattern, should
never be used, as they ultimately drop off and get lost, and the
trees are then no longer recognisable. A careful register of all the
measurements taken should be kept in something like the following
form :—

Trees.

Girth in inches on 1st January

REMARKS.

Species.

Running
number.

1890.

1893.

1896.

1899.

1902. 1905.

In the last column will be entered all pertinent remarks, such
as " Released from lateral pressure in thinnings of 1890 ;" " Over-
topped, no longer measured ;" " Found dead in 1896," and so on.
If trees of all sizes are represented in the sample plots, a series
of from 5 to 15 consecutive measurements, according to the
length of the interval at which the measurements are taken (the
shorter the interval, the larger the number of measurements re-
quired), will furnish complete data for computing the increments
for all periods and at various ages. For the jardinage type of
forests, each sample plot 'should contain' individuals of various
widely-differing ages; in all other cases, each sample plot should
contain only trees belonging to a* single age-class, or, if there is a
double storey of growth, to two age-classes.

j

C. Rate of increase of volume.

The increment put on during a given number of years « may be
obtained in several different ways as follows : —

I. BYTHB CUBING OP THE STEM IN SECTIONS. — The stem being-
cut up into sections from 6 to 12 feet long, the actual

47

and former diameters (D and d) are measured on each
section exclusive of the bark. The actual and former
contents of the stem can thus be calculated, and their
difference is the increment sought. This method is prnc-
ticable only when the annual concentric rings are distin-
guishable.

II. BY" MEASURING THE ACTUAL AND FORMER DIAMETRBS AT

HALT THE'HEIGHT OP THE TREK « YEARS AGO. — The length
of stem added on during the pastfl years is removed, and
the remaining log (whose length = /<) is then cut across
through the middle. The diameters D and d being now
measured, the increment sought

= /» = *^k (D*-d*).

III. BY MEANS OF FORM FACTORS. — In this case it is assume;!

that during short periods the form factor does not vary
in any appreciable manner. If D, If, and V denote
respectively the actual diameter, height and volume, and
d) h, and v the corresponding figures n years ago, then

This method is suitable for standing trees, when II, D and d
can be determined ; h must be estimated as accurately as

possible.

IV. BY MEANS OF THE BASAL AREA AND HEIGHT. — Let d and

I denote respectively the diameter and length or height
n years ago, and 8 and A the corresponding increments
during this interval of years. ' Then we have

r=*(d+$)* (J + A) /and* = I
Hence /„ = V— v

= * fd'l + ldU -r S8/ + d*\ + 2</A8 +8«A --<

= * f(ZdM + d^)+ (5s/ + 5W5* + 8«A) j/.

Since n is always a small interval of years, the value of the
expression in the second bracket is so insignificant that
be it may be neglected, and we have, therefore

43

+ ^A),

5 ,*/ («/ + </A)

In the last formula, when A = 0, or something very small,
then

li-

•*n ~~ */ »

0

y volume\ .

The mean annual increment cf a tree ( «ire / ls °^en as-
sumed to be the current annual increment. This as-
sumption is justifiable in the case of crops that are near
the age of exploitability, but should never be made in the
case of individual trees , the mean annual increment of
which, provided they enjoy a fair supply of ligh- on
every side, goes on increasing up to a great age.

D. Expression of the increment as a percentage.

To render what follows more easily comprehensible by the stu-
dent, we will here recapitulate briefly the general principles of
interest.

If r = the rate per cent, per annum, then I'D r - -

the amount at the end. of a year. In the case of simple interest,
we have the rule

j /Total interestX • 1

' \ Principal /— 1UU X p

Put in most cases it is necessary to assume compound interest for
which we have the following formula? :—

Amounts J = P(l-Or)«, or l-0r=//— , orr =

For the easy determination of V, Pressler has invented a formula
based on the following two assumptions':— (1), that the amount at
the middle of the period of years «• is equal to | (A + P), and

('2), that the annual increment is equal to I (A — P\. He thus

n
gets v

• r : ioo.::I (A — P): 4

A—P 200
= X —

49

This formula is only approximate, and always gives something
less than the actual value of r.

We will now apply the preceding principles to the determina-
tion of percentages for the increase of (a) the diameter, (b) the
sectional area of the stem, and (c) the volume. '

(a) Percentage of increase in diameter. — According to what

,, . D— d 200 ,.

precedes, this percentage = ^ — - x for any given section.

JJ -\- d n

The figure thus obtained cannot be applied to any other section,
since, as we know, the increment generally increases as we proceed
upwards along the stem.

(b) Percentage of increase of sectional area of ttem.— This
percentage

A—a 200

J\ *

A + a n

~

200 D'—d* 200

X— — - — V _

a O . •• r* •

»

We may also obtain this percentage (pa ) in terms of the
percentage of diametral increment ( p& ) thus —

d 100 , a 100

and -7 =

D 100-fjh A 100+*

100 a d* 100*

M onPP _^^_— — ~~ - r^* — i ^^? , -

A l>*

100 100

that is to say,

luo

a

and J"a = 2^+ g*.

In the case of rather old trees, the diameter increment of which

2

is very small, y-^- may be neglected, and then we have

'D-d 400

(c). Percentage of increase in volume.— In using method II.,
described under sub-article C. above, it is obvious that, if k it
constant, the required percentage

A— a 200 D— d 400

** ™~<

50

According to Pressler this formula gives the percentage of
increase in volume not only for the stem alone, but also for the
entire tree inclusive of the branches ; and experiment has proved
this assumption of Pressler's to be very nearly true.

In order to simplify the calculation, Pressler employs the relative

diameter A = - , in which D = the actual diameter exclusive

8

of bark, and 8 = the increment of the diameter during n years.
Now since

D = SA, and d = 8A — 8 = 8 (A — 1), we have

and also p = . _J2 x — approximately.

For the calculation of the future increment per cent., the pro-
bable diameter after n years may be assumed to be D + 8, and

. • , ! (A + l)2— A2 200.

increment per cent, to be approximately 1— — J- — -—x -- -

Pressler has published tables showing the values of this percentage
for different values of A from '2 to 300.

In the case of standing trees, of which only the increment of
diameter at the base can be measured, and rarely, if ever, the
height, the percentage of increase in volume cannot be determined
with any very jjreat accuracy. For such trees Pressler has drawn
up a set of tables for determining, with the aid of the relative
diameter, the rate of increase per cent, for five grades of height-
growth. In order to determine the percentage with sufficient
approximation without the aid of such tables, we must first find
1) and d = D — 8 by measurement, and, then the rate per centi

. ,. Tt—li 200 ,„

of increase m diameter = pd — - — ^ x - (f or a single year pd

*.JJ -J- d .11

100A\

— - — I, and the minimum increase per cent, or volume pv =

2 pd . In most instances varies from fc£ pA to 3 pd , and in the
cape of trees enjoying t ;11 height-increment and forming a
canopied crop, pv = 8|[ p* • v

In calculating the increment per cent, for a single year, we may
employ BREYMANN'S formula. On page 48 it has been shown that
tho increment on volume is approximately equal to

51

* 100 . n

so that/; = .— = 1UO ( - +

\^ hen there has been no increase in height, -

_ 200 x diauietar increment
« ' Actual diameter = ^ "

Another simple formula to use is that of SCHNLIDKK, by which

p = T-— , D being the actual diameter in inche, exclusive of the
L)n

bark, and « the nnmber of annual rings in the last inch of growth.
2
- is obviously equal to one year's increase of diameter, and this

ft

figure can consequently be used in Breymann's formula above.

Both Breymanu's and Schneider's formula give only the rate
of increase per cent, of the basal area, and the rate of increase
per cent, of volume must be obtained therefrom in different cases
as follows :—

(a) In the case of trees possessing vigorous upward growth

and standing in a leaf-canopy, by multiplying by If.

(b) In nearly all other cases of canopied trees, by multiplying

by a factor varying from li to 1|.

(c) In the case of isolated trees without any multiplication at

all in many instances.

E. Stem analysis, or the determination of the coorie of the increment of the
stem during the whole term o( it* life.

If any tree forms annual rings, the observation of these ring*
enable us to trace its life-history, to say when it was suffering from
suppression, from insect ravages, from the effects of fire, frost
and from other retarding causes, when it enjoyed complete im-
munity from such injurious influences, and was a dominant
stem growing vigorously, and' so on. A longitudinal section
passing through every point of the axis of the stem would
show all this at a glance, but such a section is practically impos-
sible to make; the stem must be cut across into a number of sec-
tions, and the required information sought thereon. The sections
may be made at every 6 feet, but as a tree growt fastest
when young, the lower sections may be longer (up to 10 and
even 14 feet in the case of quick-growing trees), while the upper
ones may be made shorter. The first cross-cut should be made at

52

the usual height of 4j feet off the ground, but there will always be
a lower one than this at the level at which the tree has been felled.

The measurement of the diameters and the counting of the rings
are operations which take time, and, will generally be best effected
in camp or at home. For this purpose a disk about 1 inch thick
should be cut off the lower end of each section, and carried away
to be studied at leisure. To prevent mistakes, the serial number
of the tree and corresponding section and other necessary informa-
tion should be noted on the upper surface of each disk as soon as
it has been cut.

The lowest section serves to determine the age* of the tree, and
the other sections the successive ages of the tree at the heights at
which they have been cut. On all of them the annual concentric
growths should be marked off in groups of 5 or 10, beginning at
the circumference. The outer diameter of each such zone (of 5 or
10 years' growth, as the case may be) should then be measured,
and recorded in the manner shown in the illustration given lower
down. The observer should be careful that the 5 or 10-year
groups, marked on the several sections, exactly correspond. Thus,
for instance, the first groups of all the several sections should con-
tain concentric rings of one and the same age ; the rings of the
second groups should be exactly 5 or 10 years younger, and so on.
To guard against error in forming the groups, extremely broad
or extremely narrow rings should be carefully noted and traced
through all the sections.

With the help of the figures obtained from the observations just
described, a longitudinal section of the stem can be delineated,
showing the whole course of growth. The scale for diameters
should be from 10 to 20 times larger than that for heights. Acen- v
tral line being drawn to represent the axis of the tree, the lengths
of the sections should be marked off on it according to the fixed
scale. At each point of division parallel lines at right angles to
the axis should then be drawn, and the measured outer diameters
of the respective concentric group-zones pricked off on them. The
required section of the tree IR completed by joining by a con-
tinuous line or curve all the points corresponding to the same age.

The rapidity of growth in tieight of the tree may be delineated
by means of -a curve, as described at page 52 of Part I of
Fernandez's Manual of Indian Sylviculture. Distances should be
marked off along a horizontal line representing periods of 5 or 10
years, as the case may be. At these divisions perpendiculars

53

should be raised, the lengths of which should correspond with the
heights successively attained at the different ages. A continuous
line joining the upper extremities of these perpendiculars will be
the required curve. The horizontal scale for years should be about
twice the vertical scale for feet.

The contents of the tree at the successive ages (every fifth or
tenth year, as the case may be) are at once ascertained from the
sectional areas enclosed by the outer boundaries of the concentric
group-zones, each sectional area being the mean sectional area of
a truncated cone, situated half in the upper half of one sect ion, and
the other half in the lower half of the one immediately above it.

Finally, the periodic increments of diameter, height and volume
can be arranged in tabular form so as to be noted at a glance.

"We will illustrate all the preceding remarks by means of an
example.

A spruce was felled at 3 inches above the ground, and the stump
showed 43 annual rings. We may assume that the tree took two
years to attain this height, and it is on record that it was sown 46
years ago. The total length of the tree was 64 feet, the stem was
cut up into lengths of 6 feet each, except the lowest portion, which
was 4 feet long. On all the sections, beginning from the circum-
ference, the annual rings were marked off into groups of five each
along three different diameters on the two lowest sections (at J and
4j feet from the ground), and along two diameters crossing each
other at right angles on the rest. For each section the mean of the
two or three diameters, according to the piece, was taken. The
results of the observations are shown in the following statement :—

' Section.

Total number of
annual rings, j

Mean diameter in ineket at Ike agt of—

:

— s

11

a

.il
w

5

10

16

20

26

30

33

40

48

h

j:

Tears.

1

2
3
4
5 '
6
7
8
9
10
11

4}
10}
16}
22}
28}
34}
40}
46}
52}
58}

43
37
33
30
26
23
19
16
12
9
6

0-51

1-42
0-39

370
255
157

5-82
4-50
3-70
2-07
0-35

753
5'92
5-40
4-20
2-52
1-39

g-ao

6-S4
6-4-1
5-59
4-31
3-33
1-83
0-59

^•79
7-68
7-36
6-69
574
5-06
391
2-59
1-01

10-17
.8-61
8-30
7-«6
6*98
r, »•_•
5-61
4-53
3-10
1-88
>::•

12-40
940
9-03
- II
7 v
736
6-68
6-87
4-67
364
1-88

12-87
g u

9*43

x-x,»
s -J.i

7-74
7-01
I -JJ

m

:< x,;
2-10

CALCULATION OF VOLUMES ACQUIRED SUCCESSIVELY AT 5, 10, 15 45 YFAUS or AGE.

•^Fq qjiM
ucal et »« )93j
ejcnbs ni vajy

-^00:e<M«OOOTC'*iCi<N
• OiCOOi— it-t—OOSOi— ICO
iMOO^t-tMCDOCOCOlM

lOTji-fuoec^iiMi— 100

SO O ^O
r— 1 t-» r— 1

? 55 66
«

04

00

t~

1— 1

c6c6666666

i

1

•«
•S

«

IS-
IS

1

>)

§

E

£

-*OOCO2<lCOOS«OOfr-(N

ost-iocaojowoocii— ios

l-»^X«CCl'#QOi-it»— t

T-fTjtcoeojSNfir-lOO

(M (N »0

t>. US O

S 3 ° :"

<N

to

CD
i— 1

o o o o o o o 6 o o

O

t^OiCCiOOtO^COC^
iOCC(NS<ICO(MOC<:Oi
<_'*>. S<l CO (M t^ rH 10 r-< :
•^"COCOOHSMfli-iOO •

§ S 3

« a b ;
i— i

00
f— 1

71
t— t

oooooooco

8

CC?DC5CQOO^>OOU?
1MCOCCOO5^«5*O

NO-^IQOWIOOCCO : :

CQ:N<Mi— ^-iCOO • •

oo us t-i

C» QO -^t

^ »- o •

1— 1

«o

<M

00

60660660

8

— O CO •<# 00 OS
CO SO OO f-l O5 !>.

ion«oo>opH : : : :

!M (jq ,_ _ o O • • ' •

O CO (M Tf
t- Ol «C O
<M ^ O

oo "* <=o
o

oo

9

»b

0 0 O O O O

M

1C ® CD — 05
7-1 *>. 10 rj< O

Oi ^ "^ *^
C> OS <N O

$ <* ^9

6

00

i— i

CO

a

^* iJ>J 5<1

o -^ oo
i— i t>i IN : : : • :• : :

00 «O «O OO

1 2 S|

o

1— 1

"?

r-l

r+ O O

666

IO

00 «O
US 03

1—1 OS IO TO
OS <?1 O i— I

| 0 6g

«3
09

6

o o : :

6 6

2

CD

: : :3

• 6

co
g
6

•a

rH

: : :8
•6

i— i

8

6

•jaaj ui noijMs
jo W*PH

•<#OCOiN t> -+ ~ «O iM 00
^i-((N'NpO^^»OW3

Total area in square feet of sections of the same age .
Contents in cubic feet of the 6-foot lengths at the succes-
sive ages . . . . . ... .
Contents in cubic feet of the lowest 1£ foot length at the
successive ages ........
Contents in cubic feet of top end at the successive ages

Total contents in cubic feet of tree at the successive ages

V

1

4

X

56

TABLE SHOWING COURSE OP INCREMENT OF THE STEM.

Ape in

Diame-
ter in
inches.

JTfiffkt
in

feet.

1 ftlutfif in
cubic feet.

Form-factor.

I iii-mii i-ii t of
vnluin- per
oeut.

ye»

"3

a

Z
£

-a

•8

Ol

a

£

1

8
B

E

J

"a

A-/f

lOp-^v

o

R

3

8

|

o

^

1

<

5

2

o-ooi

5

0-025

10

04

7

0-026

22

9

032

15

2-6

16

0-35

19

8

1-06

**

24-0

20

4-5

24

141

0'529

0-472

1-4

9

1-77

1V3

25

5-9

33

3-18

0-490

0-445

0-9

9

2-10

10"0

30

6-8

42

5-28

0-499

0-462

0-9

8

2-98

8-6

35

7-7

5C

8-26

0-513

0-481

09

9

8-92

76

40

8-6

59

12-18

0-513

0-489

0-8

5

399

6-8

45

9-4

61

16-17

0-530

0-504

Inclusive of >
burk, j

9-8

64

17-82

0-63P

~

5. Kate of increase of crop*.

lu the case of crops it is obvious that, we require to know only
the rate at which the volume of the crop increases, and that
a knowledge of the rates at which the height of the crop and the
aggregate diameters and basal areas of the component stems in-
crease, is of interest only so far a& these element* are so many
factors in the growth of the crop.

A. Determination of the increment ftt nny lime.

There are various methods of determini-ng the rate at which a
crop has been augmenting in volume at any time. They are —

I. BY MEANS OP INVESTIGATIONS IN THE CROP ITSELF. — In

determining- the increment it must never be forgotten
that no oue class of stems can be taken as representative
of the whole crop, any more than any individual stem

56

can be taken as representative of its class. It must also
be borne in mind that so far as the standing timber alone
is concerned, thinnings cause the rate of increase to fall.
Lastly, we have the fact that what is now an average
stem of the crop was at one time an overtopping or domi-
nant tree, and may in the future become a dominated or
overtopped or even a suppressed one.

The simplest way of determining the increment of a crop with
the nearest approximation is to determine first the per-
centage of increase of several stems of each and every
component class. If the crop is at the time under a
regular valuation survey, this percentage is calculated for
the section passing through the middle of the felled
sample trees, and the data thus obtained should also be
supplemented by measurements made on standing trees.
If there is no valuation survey going on, then measure-
ments made on standing trees must supply all the required
data.

If Flt F2, ............ have been ascertained to be respectively

the volumes of the several stem-classes, and plf p^, ......

...their average increment per cent., then the increment

V 01-
for one year (i) of the several classes will be, il = • '

V <o
ia = 2 Q- ........ , .......... and the total increment of the

crop

= Ie = t\ + *2 ............... * -,

so that the increment per cent, of the crop will be
1001

;* r^+r, +...,.

The formula just investigated should be used only when it is
required to know the increment for a single year, or for a
short period. If it is sought to determine the increment
for a long period of w- years, another procedure must be
adopted. pi} /?2,....^.being found to be the percentages

of increase of the respective stem-classes during this period,
the volumes vlt w2, ....... ..of these classes « years ago

may be determined from Pressler's formula thus—

-

57

The increment for the n years will hence be —

TD = (P, + Fa + )-(*, + *2 + ).

Lastly, if it is required to ascertain the probable increment
for the next n years, we must determine the probabU
volumes F/, Fa',...of the several ste'm-ctasses at the end
of n years by the same formula of Pressler's
r/

1
The increment for the » years will be

IB' = (v\ + i\ )- (!\ + F2 )= r.-r.

The rate per cent, of increase for the same period will be —

, r. + r. 220.
P» '• '• r\ — vt "V

From all the preceding expressions it will be obvious that the
increment per cent, of the whole crop can be equal to
the arithmetical mean of the percentages of increase of

the sample stems only when Ft — V^ = F3 ; and

this mean will be approximately correct if the cumber of
sample stems is very large and they are mostly taken from
the dominant class.

II. "WlTH THB AID OF PERCENTAGES THAT HAVE BEEN PRE-
VIOUSLY OBTAINED FROM ACTUAL INVESTIGATIONS IN SIMILAR

CROPS. — In this case three facts have to be borne in mind,
viz., (\) that the percentage of increase falls as the age of
the crop increases, and can be assumed as constant only for
short periods; (2) that the percentage falls niore rapidly,
the more quickly the individual trees increase iu diumete1*
and volume (crops of poor upward growth yield a higher
percentage than vigorous crops of equal age) ; and (3)
that the lighter crop has a larger percentage than the
denser one.

This method is employed in open crops and coppice standards in
neither of which cases are tables of yield directly applicable.

III. WITH THE AID OP TABLES OF YIELD. — When such tablet art

used, the locality, density and age of the crop in question
and of the crops from which the tables have been
calculated must correspond very closely.

IV. BY ASSUMING THE INCREMENT BOUGHT TO BE TUB MIAN

ANNUAL INCREMENT AT THE PRESENT AGB OP TH1 CROP.—

This method is applicable to exploitable crops.

58

B. Determination of the moan increment of a crop for the entire period 111
which it reaches exploitability.

In the case of all crops that are nearly exploitable the mean in-
crement at the present age is for all practical purposes the mean
increment sought. It is only in very old crops that the present
mean increment will be found to be too small. For very young-
crops, and in crops much younger than the age of exploitability,
the required mean increment must be obtained from yield tables.

C. Course of the increment of a crop during the whole period of its life.

The stem-analysis (see page 53) of the sample stems can give
the course of growth solely of the present main crop (i.e., the crop
exclusive of the overtopping, overtopped and suppressed stems), and
as regards the crop at previous ages it can furnish only the volume
and height of the largest trees which the crop then contained.
Hence, to be able to trace the course of growth of a crop through
its successive ages, the same crop must be successively surveyed at
each of those ages.

D. Which method of determining the increment of crops to employ according
to the purpose for which the information is required.

The determination of the increment of entire crops is generally
undertaken for the purpose of framing working-plans, and the ob-
ject then is to ascertain either what the yield of the several crops
will be when they are felled, or at what respective ages they will
severally become exploitable. v

"When it is required to determine the mean annual increment of
a crop up to the time it becomes exploitable, yield tables should be
used if the crop is young, and the formula ^eSsnSr" ^ the cr°P \
is nearly exploitable.

When it is required to know what a crop will yield if it is felled
within the next 20 years, we must add to the present volume the
probable increment for the ensuing 5 or 10 or even 15 years. This
increment may be determined either (1) as a percentage in the crop
itself, or (2) by adopting such percentages as usually accrue in
similar crops at the age of the highest mean increment, or (3) by
adopting the mean annual increment of the given crop at the pre-
sent age, or (4) by using the increment furnished by yield tables.

In some methods of framing working-plans it is necessary to
know what the yield at the time of felling will be of all the crops,

59

from the youngest to the oldest. This information can be obtained
only with the help of yield tables.

Whether a given crop is exploitable or not must be determined
by the fact whether the percentage of increase demanded 10 being
produced by the crop. This can be settled only by investigation*
in the crop itself.

4. General remarks on the count? of rtertlnfiment of the imliti-

dual tree*
The following remarks apply only to seedling trees and not to

coppice shoots : —

A. Growth in height.

The growth in height is at first nearly always very slight, and

in India remains so, according to the species and to the soil and

locality, for a period extending from 3 to 10 and even 15 and up

to 20 years. During this time the seedling is technically raid to

be establishing itself. As soon as the seedling is thoroughly e*.

tablished, the rate of growth in height increases rapidly, and

attains an annual maximum in a comparatively short time. In

Europe this maximum is attained in the case of pines and larch in

10-15 years, in the case of the spruce in 20-25 years, and in

the case of the beech and silver fir in 30 years. For India we have

unfortunately no exact figures, and owing to the continental

variety of its soils and climates, one and the same species presents

extremely wide divergences. The maximum rate of growth only

lasts a short time. The rate sinks rapidly in the case of species

in which the maximum is attained early, more slowly in others,

until it is reduced to from 3 to 6 inches a year, at which figure

it keeps for a great number of years. A total cessation of growth

in height occurs only at a very advanced nge, and earliest in wo.

lated trees.

Rapidity of upward development reaches it* maximum earliest
and begins to fall quickest in the most favourable soils and locali-
ties. In unfavourable localities, as on mountain rid«res, the rate
at which a tree grows up remains nearly constant during its whole
life, after it has once attained a certain figure. In youth and
middle age the rapidity of growth in height in the mo*t favourable
localities exceeds greatly that in unfavourable ones ; afterward*
there is but little difference. The height reached by tree* in
mature crops is from two to three times as great in favourable!**

GO

lities as in unfavourable ones. It has been recently established that
a close leaf-canopy not only checks growth in girth but also
growth in height, although the latter is not influenced by the
density of the crop to the same extent as the former, and the un-
favourable influence begins to sho'w itself only in very dense
crops.

B. Growth of girth and basal area.

The course of growth of the girth at ground level is similar
to that of the height of the tree. The girth is, however,
usually measured at breast-height (4j feet). Hence a curve de-
lineating its growth cannot start from zero, but from the point
of time at which that height was attained. By the time a tree
reaches this stage, the rate of growth of the girth has either
entered upon its maximum or is on the point of doing so. There-
fore, as we are accustomed to measure it, the girth starts at
or near its maximum, rate of increase. When the rate of growth
begins to decline, it does so, at first more or less rapidly according
to species, soil and locality, slowly afterwards, and remains more
or less constant for some time.

The rate of increase of the basal area is very slight at first,
then augments more or less rapidly up to a certain figure, after
which it remains constant or diminishes slowly. The continuance
of a dense leaf-canopy up to an advanced age results in an early
and rapid decrease of the rate of growth of the basal area of the
individual component trees. On the other hand, in the case of trees
standing isolated, the rate of growth increases, or at least remains
constant, beyond even the ordinary age of exploitability. In the
most favourable soils and localities the rate of .increase of basal area
Of the individual tree attains its maximum. rapidly (about the 40th V
or 50th year for European trees) and then declines; whereas under
opposite conditions it augments slowly, but the augmentation con-
tinues up to a great age.

C. Developme'ht of the form -factor.

The development of the form-factor of a tree is dependent on
the rate of increase of the girtfy at different heights, which rate itself
depends on the amount of standing room and the consequent expan-
sion of the crown. In the case of trees forming a leaf -canopy the
width of the annual rings of growth is generally greatest at the

61

top, and diminishes downwards to some point close to the ground,
and then increases again down to the crown of the roots. The
rings are thus narrowest at a certain point in the lower part of the
stem, which point is almost on a level with the ground in young*
trees, but gradually moves upwards with increasing age and the
formation of buttresses. In exploitable trees this point is generally
situated above the usual height of measuring the diameter, vit.t 4j
feet; and in very old trees, and in those which possess free grow-
ing-room, it is found as higli as 12 to 24 feet above the ground.

The increasing width of the annual rings from bottom up-
wards is most marked in favourable soils and localities and in trees
in the midst of a dense leaf-canopy (t>., trees with a long bole
and a small high crown), and least so in unfavourable soils and
localities and in overtopping and, especially, isolated trees. It is also
most conspicuous in trees that are growing up vigorously, and this
particularly in the upper part of the stem, whereas, on the contrary
in trees pushing up slowly or which have entirely ceased to grow,
the width of the rings diminishes again towards the top. In the case
of isolated trees with low-spreading branches the rings are of the
same width throughout the entire length of the stem, or may even
become narrower from bottom upwards.

Young trees have an absolute form-factor of from 0*30 to 0'85.
With increasing age these figures rise to 0'44 and even 0'48 ; but
they ultimately diminish after an advanced age is reached. This
decrease occurs in the European larch at the age of 80 — 100 years,
and even earlier in trees grown out in the open. Trees that have
developed in isolation always have a low form-factor. In favour-
able soils and localities the form-factor is higher than in unfavour-
able ones.

The rate at which the stem expands at different heights is ob-
viously not the same as that at which the girth increases. The
increment of sectional area is greatest at the level of the soil,
decreases rapidly upwards for a short distance, then much more
gradually up to the beginning of the crown (sometimes even in-
creasing in the vicinity of the crown), and Instly, diminishes very
rapidly upwards to the top of the crown, where it consists merely
of the sectional area of the previous season's shoot. In trees
growing in the midst of a dense leaf-canopy, and in those already
dominated, the largest increment of sectional area occurs in the

02

upper part of the bole ; whereas in isolated trees it is to be found
much lower down. In canopied crops growing in favourable soils
and localities the increment of sectional area is almost the same
throughout the entire length of the stem ; iu unfavourable soils and
localities and iu the case of all isolated trees, whatever the nature
of the soil and locality, it steadily decreases from below upwards.
D. Rate of increase of volume.

Increased mass is the result of the co-operation of three factors —
increase of height, increase of girth and augmentation of the
form-factor. In early youth, in spite of the great width of the
concentric rings of woody growth, the increment of volume is
small j it reaches an important figure only when the crown has
acquired some development, and the stem, by its increased height
and girth, presents a sufficiently extended surface for the deposit
of new woody growth. After this period the rate at which the
volume increases rises rapidly to its maximum. In Europe this
maximum is attained at the age of 50 — 70 years in the case of
quick-growing species in suitable soils and localities; at the
age of 100— 120 years in the case of slow-growing species under
unfavourable conditions ; and at a very advanced age by trees
standing out in the open, or situated on high exposed ridges or at
great elevations. Once at its maximum, the annual rate of increase
remains more or less steady for a long time, after which it declines,
but at a less rapid rate than that at which it rose.

The mean annual increment of the individual tree, as a rule,
attains its maximum only at a very advanced age, generally beyond
that of ordinary exploitability. Even in canopied crops the maxi-
mum is not reached by the dominant and over-topping trees before
the age of 120 — 140 years.' Trees growing out in the open, and
individuals of species which develop "slowly during their youth,
attain it much later; while in high mountainous regions many trees
as much as 300 years old may be found which have not yet entered
upon that stage.

5. General remarks on the growth of the crop.

The course of development and the accretion of volume in timber

crops depends not only on the species, soil and locality, but also on

the treatment and system of working' adopted. In respect of one

and the same species the amount of production is influenced chiefly

63

by the soil and locality, the course of development of the crop bj
the treatment and system of working1. Of the course of growth in
crops worked by the jurdinage and coppice methods but little it as
yet known, bat a flood of light has been thrown on the growth of
regular crops treated by the uniform method l«y the lubotiic of the
German Department for Forest Research. The remarks which
follow refer only to such crops.

A. Natural constitution of utem

In every crop we can recognise, besides the over-topping
dominant individual, two lower classes of dominated and over.
topped ones, nnd, if no thinnings have been made, also a tilth class
of suppressed stems. The first two classes form what mav be
called the main crop, the other three the subordinate crop. It in
obvious that in the ordinary course of development of the crop
fresh stems are constantly passing into the lower classes from the
immediately upper ones, and from the main crop into the subordi-
nate one, from which they are ultimately removed by thinnings or
natural decay and death. The result is a constant diminution of
the number of stems composing the main crop.

The original number of individuals in a crop depends on the
manner of its constitution, according as it has sprung up from
self-sown seedlings or from artificial sowing (generally executed
close), or from transplants (generally put out comparatively far
apart). The number of stems diminishes rapidly in youth, lo.-s
mpidly in middle age, and still more slowly in old age; rapidly in
favourable soils and localities ; slowly, but steadily, up to a great age
in unfavourable places. At. one and the same age more stems stand
in unfavourable soils and localities than in those more suitable.
A crop of Scotch pine of the best quality contains 1,200 stems per
acre at 30 years of age, but only 140 stems 90 years later at the
age of 120 years.

This constant and great diminution in the number of component
individuals results in a considerable falling off of the increment,
the consequence being that both the current and mean annual in-
crements reach their culminating point earlier in the caro of the
crop th:m in that of the individual belonging to the dominant or
representative class.

Since thinnings and decay and death remove mostly the iudivi-

64

duals of the lowest class, the average stem of the crop is constantly
moving upwards into one of the (up to the present) larger stem-
classes" and on the other hand the average stem at any given age
is constantly receding into a smaller class and ultimately takes its
place in the subordinate crop. So that the exploitable crop eventu-
ally consists for the most part of individuals which in their youth
belonged to the highest or over- topping class. Hence investiga-
tions into the course of growth of a crop, by means of measurements
and ring-countings made on existing already exploitable indivi-
duals, give the heights, diameters, basal areas, volumes, etc., at
various periods, not of the representative or average individuals at
those periods, but of the largest class of stems of those periods.

B. Basal area.

According to the universal convention adopted of measuring the
diameter at breast-height (4j feet), the basal area of crop is
obviously nil until that height is attained ; it then increases
rapidly up to middle age, and thenceforward more slowly but
steadily up to a great age. The diminution of the rate of increase
is very conspicuous in the case of quick-growing or shade-avoiding
species, but is comparatively slight in the case of shade-enduring
or slow-growing species. Thus the basal area in good crops of the
European spruce or silver fir, at the age of 140 — 150 years aggre-
gates as much as 348 square feet per acre, whereas in the best
crops of Scotch pine or beech it seldom exceeds 217 square feet per
acre. In inferior soils and localities the b^sal area, for one and the
same age, is considerably less than in good soils and localities in
spite of the number of stems being larger. In Germany, in un-
favourable localities, the bas,al area in mature crops is only 130
square feet per acre for pine and larch, and 196-1— 21 7 square feet per
acre for spruce and silver fir. In Germany the basal area of a crop is
on an average about 0*5 per cent, of the area covered by the crop ;
in the best soils and localities the average percentage is 0*8 for
spruce and silver fir.

C. Volume.

The volume of a crop, as well as the rate at wtrich it increases, is
very sm:ill in early youth. The volume then increases rapidly up
to the end of the middle age of the crop, after which the rate «i
increase is much less rapid, but is maintained up to a very advanced

C6

period. A complete cessation of increase can occur only wlien the
increment of tbe growing stems is just counterbalanced l.y the )OM
due to decay and death.

The current increment of volume becomes rapidly larger until
it attains its maximum in 3<> — 40 years in the case of quick-growing
species under favourable conditions, and in 7.0 — 80 years in the CAM
of slow-growing species and in unfavourable aoilg and localities.
After reaching its culminating point, it sinks rapidly in placet
where favourable conditions exist, more slowly where circumstances
are not so suitable.

The mean annual increment obviously begins by being identical
with the increment of the first year. As the current increment
now goes on increasing steadily year by year, as long as this in-
crease continues, and also for some time afterwards, the mean in-
crement obviously keeps below the figure of the current increment.
Ultimately it catches up and gets ahead of the latter, and as the
current increment goes on steadily declining, the mean increment
maintains its superiority to the end. The mean annual increment
obviously attains its maximum when it becomes equal to the current
annual increment, for the continued diminution of the latter must
cause it to decline also from that point. As the current increment
diminishes only gradually, especially in unsuitable soils and locali-
ties, the mean annual increment, after attaining its culminating
point, continues nearly at the same level for a considerable period,
particularly so in unfavourable places, where this period may extend
over several decades.

Chapter VII.

On the compilation of tables of yield.

In yield tables are collected figures representing the coarse of
growth of different classes of crops that have developed under nor-
mal conditions of growth and density. These figures give the
volume and increment per unit' of area of the crops in question at
different ages and under different conditions of growth, and some-
times also the corresponding factors which contribute to the pro-
Auction of volume, viz.t number of stems, basal area, and height
of crop.

As shown higher up, the course of development of a crop can-
not, like that of the individual tree, be deduced by a single i*rie«

B

66

of investigations made at the age of exploitability. The number
of stems, the basal area, and the dimensions of the average stem
at all previous periods must be known. Hence to trace the course
of growth of a given crop it must be surveyed from its earliest
youtb annually, or at regularly recurring periods, until it becomes
exploitable. This metliod of repeated survey has been adopted in
order to measure the influence of different modes of treatment on
crops that in all other respects are similar.

To follow out this system rigidly on one and the same crop
through the whole of its life would be an extremely long and slow
process. In order to curtail it and obtain the results sought as
quickly as possible, several similar crops, under the same treatment
but of different ages, are experimented upon, and the various
observations made from time to time in the several crops are com-
bined and interpolated together into a connected whole, that is
nearly, if not quite, as accurate as if the figures it comprises had
been obtained from investigations in one and the same crop from
the time of its constitution to its maturity.

There is yet another and still shorter way of obtaining the re-
quisite figures. A series of crops growing under similar conditions
of species, soil, locality and treatment, but of various ages differing
from each other by as short intervals as possible, is chosen with
great discrimination and care, and each crop is measured once for
all, the results being tabulated together. Most of the yield-tables
hitherto compiled have been obtained by this method, and the
results have been proved to be quite correct enough to justify its
adoption as often as there is no time to wait for the outcome of the
longer and more elaborate methods.

Whichever of the two Idst described methods is adopted, there
are always certain ages for which figures are wanting, and the
gaps must be filled up by interpolations which are most conveni-
ently obtained graphically thus : — The various successive ages are
marked on a horizontal line, and at these points perpendiculars are
raised, the perpendiculars for the ages for which the volumes or
increments are known being made of the lengths corresponding
ia these volumes or increments) as the case may be. The ends of
these perpendiculars being joined by a continuous curve, the
lengths of the other perpendicular up to where they are cut by
the curve, give respectively the remaining volumes or increments.

67

rJ he figures obtained for the volumes should be checked by con«
structing similar curves with the known numbers of trees, Lasnl
areas and heights, and obtaining by interpolation the corresfionding
figures for the oth.er ages. The complete figures obtained for each
series should then be compaied : wherever discrepancies are found
corrections can be easily ncade. Such a check for the volumes is
best obtained by comparison with the figures obtained for the ba-al
areas and heights, the products of which, multiplied by the respec-
tive form-factors, will furnish another series of volume figures.
The form-factors used may be obtained at the same time as the
other information.

The accuracy of yield tables is always more or less uncertain,
owing to the difficulty of selecting the experimental crops BO that
they may be exact counterparts of each other at their own respec-
tive ages. The necessary correspondence is best secured by
careful and detailed stem analyses in the older crops, in order that
the heights of the dominant stems at earlier periods may be more
or less exactly a&certained. The experimental crops for these
earlier periods should then be so selected that their average heights
correspond with the figures thus deduced.

Government of lu<li» Central Printiu.- OfHc«.-Ko. lOa? B. 1 A.-lA-S-M.-l.tMb— J. K.U. feC

LIBRARY

FACULTY OF FORESTRY
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Carter, Philip John
555 A treatise on the
C37 mensuration of timber and

timber crops

Forestry

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