Forest stocking equations : their development and application

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, APR JUJ973

FOREST STOCKING
EQUATIONS:
Their Development
and Application

by Peter F. Ffolliott
and David P. Worley

March 1973

USDA Forest Service
Research Paper RM-102
Rocky Mountain Forest
and Range Experiment Station
Forest Service

U.S. Department of Agriculture

Abstract

Using point-sampling techniques, stocking conditions at a
sample point can be described in terms of whether or not the point
is stocked to a minimum basal area level corresponding to a
particular basal area factor (BAF). Stocking equations relating
proportions of a forest stocked to minimum basal area levels
corresponding to each BAF used in an inventory can be defined by
regression analyses. Stocking equations can be used to help
evaluate land treatment potential, determine treatment feasibility
on a single management unit, and as a basis for setting
operating priorities on a number of management units.

Keywords: Stand density, basal area measurement, forest
management, forest surveys.

USDA Forest Service
Research Paper RM-102

March 1973

FOREST STOCKING EQUATIONS:
Their Development and Application

by

Peter F. Ffolliott, Associate Silviculturist

and

David P. Worley, Principal Economist
Rocky Mountain Forest and Range Experiment Station^

'Research reported here was conducted at the Station's Research Work
Unit located at Flagstaff, in cooperation with Northern Arizona University;
Station's central headquarters maintained at Fort Collins, in cooperation with
Colorado State University. Ffolliott is currently associate professor. Department
of Watershed Management, University of Arizona, Tucson; Worley is with USDA
Forest Service Northeastern Forest and Range Experiment Station's Unit at
Columbus, Ohio.

Contents

Page

Introduction 1

Theory Behind Development of Stocking Equations 1

Synthesis of Stocking Equations — An Illustration 2

Study Areas 2

Methods 2

Results 3

Applications of Stocking Equations 5

Setting Realistic Limits for Forestry Practices 5

Decisionmaking at Different Levels of Interest 6

Setting Operating Priorities 6

Development of Distribution and Density Functions.... 6

Summary and Conclusions 7

Literature Cited 8

^FOREST STOCKING EQUATIONS:

Their Development and Application

Peter F. Ffolliott and David P. Worley

Introduction

Point sampling techniques are widely
used for inventorying forests. These inventories
provide data on average basal area (or volume,
number of trees, and so forth) per acre for forest
managers. Such inventories may yield
additional information regarding the propor-
tions of a forest stand stocked to minimum basal
area levels.

Even-aged stands with regular spacing
patterns resulting from plantations or extended
periods of management can, possibly, be
described by average basal area per acre. With
uneven-aged stands of irregular spacing
patterns, however, another statistic — the
proportion of the stand stocked to a minimum
basal area level — would be useful to set realistic
limits for forestry practices, judge suitability of
an area for management practices, or set
priorities for cultural or harvesting operations
among forest areas. Such information can be
derived from stocking equations, as described
here.

The purposes of this paper are to: (1)
outline the theory behind the development of
stocking equations, (2) illustrate the
methodology of stocking equation synthesis,
and (3) demonstrate applications of stocking
equations in forest management decision-
making.

Theory Behind Development of Stocking
Equations

The basic theory of point sampling is well
known. The number of trees tallied at a sample
point, multiplied by the basal area factor (BAF)
used, gives an estimate of basal area per acre at
that sample point. A sample point is stocked to
basal area levels of 50, 70, and 100 square feet per
acre on the basis of 5, 7, or 10 trees tallied with a
BAF of 10. Estimates from a number of sample
points are averaged to estimate the basal area of
the forest area.

The use of a single BAF can incorrectly
describe the stocking situation at a single point,
however, due to irregular spacing patterns and
the variety of tree sizes frequently associated
with natural timber stands (fig. 1). For example,
a sample point would be considered stocked to a
basal area level of 75 square feet if three trees
were tallied with a BAF of 25. But, it is possible
that no trees would be tallied with a BAF of 75, in
which case the sample point would not be
considered stocked to 75 square feet. Conversely,
assume one of the three trees tallied with a BAF
of 25 is close enough to the sample point to be
tallied with a BAF of 100. The tally with a BAF of
25 would underestimate this stocking condition.

Empirical trials in cutover ponderosa pine
stands in Arizona corroborate the above
examples. Sample points with a single tree

Figure 1. — Irregular spacing
patterns and intermixed
size classes in Arizona
ponderosa pine stands.

2*-l

tallied with a BAF of 25 were also stocked with a
BAF of 50 and 75 half the time. Sample points
stocked with two trees with a BAF of 25 were
stocked, with a BAF of 50, only 84 percent of the
time, and sample points stocked with three trees
with a BAF of 25 were stocked with a BAF of 75
only three-quarters of the time.

If it is not possible to correctly describe
stocking conditions at a single sample point with
a single BAF, it follows that it may not be
possible to describe the proportion of a forest
stand stocked to arbitrarily specified basal area
levels. A more accurate method for determining
the proportion of a stand stocked to different
basal area criteria is through the use of stocking
equations. The synthesis of stocking equations
is based on two assumptions.

First, a sample point is considered stocked
to a given minimum basal area level if at least
one tree is tallied with a BAF corresponding to
that level, or not stocked at that level if no trees
are tallied. This concept about stocking
conditions has been suggested as a way of
determining the proportion of a stand stocked to
a single minimum basal area level defined by a
specified management objective (Roberts 1964).
In the absence of specific management guide-
lines, an inventory system employing a range of
BAF's, allowing sample points to be described in
terms of being stocked to a corresponding range
of minimum basal area levels, is advantageous.

Secondly, the proportion of a forest stand
stocked to a given minimum basal area level can
be estimated from the proportion of sample
points stocked to that minimum level, provided
the sampling of stocking conditions was
unbiased. Similarly, relationships can be
established between proportions of a stand
stocked to minimum basal area levels
corresponding to each BAF used in the
inventory. These relationships assume mathe-
matical forms which can be defined empirically
through regression analyses. The equations
describing these regressions are stocking
equations, the dependent variable being the
proportion of a forest stand stocked to minimum
basal area levels within limits dictated by the
BAF's used in the inventory.

Synthesis of Stocking Equations —
An Illustration

To illustrate methodology, stocking
equations have been developed to describe
cutover and virgin ponderosa pine (Pinus
ponderosa Laws.) stands in north-central
Arizona.

Study Areas

Data representing a cutover ponderosa
pine stand were collected from watershed 12,
encompassing 425 acres, on the Beaver Creek
watershed (Brown 1971), 45 miles south of
Flagstaff. At the time of measurement, half of
the merchantable sawtimber volume had been
cut from the area between 1943 and 1950.
Sawtimber volume averaged 3,700 board feet per
acre, and the site index (Meyer 1961) varied from
45 to 60 feet at 100 years. Soils, derived from
basalt parent material, are classified in the
Brolliar and Siesta-Sponseller soil management
areas (Williams and Anderson 1967). The area
was sampled with 197 points arranged in four
random starts with four strata (Shiue 1960).

Data from a virgin ponderosa pine stand
were collected on the Long Valley Experimental
Forest, 65 miles southeast of Flagstaff. At the
time of the study, this was one of the few
remaining areas in Arizona where a virgin stand
could still be found on a good timber-growing
site. Timber on the 1,280 acres comprising the
Forest was uneven-aged, with different age
classes occurring as small, even-aged groups.
Sawtimber volume averaged 20,500 board feet
per acre, and the site index (Meyer 1961) was 85
to 90 feet. Soils, formed from limestone and sand-
stone, are classified in the Hogg-McVickers
series (Anderson et al. 1963). One hundred sixty-
six points arranged in an 8-chain by 8-chain grid
provided the sample design here.

Methods

Sample points on both study areas were
considered stocked or not stocked on the basis of
trees tallied with an angle gage corresponding to
BAF's of 5, 10, 25, 50, 75, 100, 125, 150, 175, 200,
and 250. Diameters (d.b.h.,o.b.) of all tallied trees
were recorded to allow subsequent assignment
into size classes.

The data were subjected to regression
analyses to develop stocking equations
describing (a) all size classes and (b) individual
size classes on the two study areas. Linear
regressions had been used previously to describe
the proportions of a cutover ponderosa pine
stand stocked to minimum basal area levels
between 25 and 75 square feet (Ffolliott and
Worley 1965). Here, straight-line prediction
mechanisms performed well over a limited range
of basal area levels. More complex curvilinear
forms were required, however, for regressions
that defined stocking conditions near the
extremes of a stand population — low or high
stocking levels for all timber size class elements
within a stand — or for particular size-class

2

components. Furthermore, several scatter
diagrams revealed an inflection point near the Y-
axis. Consequently, commonly used linear trans-
formations proved unsatisfactory.

A computer program (Jameson 1967) that
approximates a 5-parameter transition growth
curve describing general sigmoidal relation-
ships (Grosenbaugh 1965) was arbitrarily
selected to illustrate the development of stocking
equations. Other mathematical models and
computer programs designed to produce a
sigmoid form might give similarly good results,
j As a practical matter, the closeness of fit to the
data illustrated in figure 2 suggests hand
plotting may be adequate for developing curves
for many purposes.

Results

Stocking equations describing all size
classes of ponderosa pine on the two study areas
are illustrated in figure 2. Stocking equations
developed for the sawtimber (at least 11.0 inches
diameter), pole (4.0 to 10.9 inches diameter), and
sapling (less than 4.0 inches diameter) size
classes (fig. 3) are summarized as follows:

Beaver Creek (watershed 12)

(1) Sawtimber

Y = 100 - 89.1 (l-e-°-"'")

(2) Poles

Y = 100 - 89.2 (l-e-°-°i^") "'^

(3) Saplings

Y = 100 - 103.8 (1-6-°'°°^'^) °-
Long Valley Experimental Forest

(1) Sawtimber

Y = 100 - 24.1 {■[-e-'-°°'n

(2) Poles

Y = 100 - 96:5 (i-e-°-°°«'^) "'^

(3) Saplings

Y = 100 - 101.1 (1-6-°-°'^'^) °-

3

Summation of the three size-class components at
a given basal area level may exceed the stand
population stocking at that level, since many
sample points were stocked with more than a
single size class.

The stocking equations allow us to
estimate the proportion of a forest stand stocked
by a stand element to minimum basal area levels
up to 250 square feet. Solving stocking equations
for numerous alternative basal area levels may

4

i

become time consuming. To ease computations,
a supplementary computer program can be
written to solve equations in terms of the
proportions of a stand stocked to any inter-
mediate basal area. Generally, estimates
obtained from a graphical presentation of the
stocking equation will suffice.

Applications of Stocking Equations

A truly adequate description of the char-
acteristics of timber on a management unit
must answer a variety of questions of manage-
ment specialists regarding timber production.
Stocking equations help provide such answers,
by defining the proportion of a forest stand on a
management unit stocked to minimum basal
area levels dictated by management objectives.

Setting Realistic Limits
for Forestry Practices

Stocking equations can help a manager
reach a decision as to the feasibility of imposing
a treatment (such as harvesting, thinning, and
so forth) on a management unit. It is assumed

that the proportion of a forest stand stocked to a
minimum basal area level which corresponds to
the basal area level prescribed by treatment will,
subsequently, represent the proportion of the
stand that will be placed under treatment.

For example, suppose a silvicultural
practice calls for a uniform thinning of all
sawtimber in a forest stand to a basal area level
of 50 square feet per acre, the assumed
"optimum" in terms of a sawtimber manage-
ment potential. However, a stocking equation
developed for the management unit may reveal
only 43 percent of the stand could meet the treat-
ment stocking objective (fig. 4). A decision may
then need to be made regarding treatment
feasibility. Possibly, the original prescription
could be discarded in favor of one that would
place a larger proportion of the stand on the
management unit under treatment. This could
be achieved by reducing the uniform thinning
treatment to 25 square feet per acre. Unfor-
tunately, thinning to this alternative stocking
level may result in a lower sawtimber manage-
ment potential. Due to the greater proportion of
the area treated (fig. 4), however, the outcome
could be more favorable in the long run. The final
decision must be a compromise between
obtaining the maximum management potential,

_ 100 -

Figure 4. — Graphic representation of stocking
equations describing sawtimber size class
in an Arizona ponderosa pine stand (Beaver
Creek watershed 12). Dashed lines refer
to a text example of the application of
stocking equations.

50 100 150 200
Minimum basal area (sq.ft. per acre)

250

5

as prescribed by treatment, and extending the
treatment to the largest possible proportion of
the stand on the management unit.

Regardless of what a specific land treat-
ment is to accomplish, the application of
stocking equations will help evaluate treatment
potential and prescribe treatment feasibility. A
range specialist may ask "What proportion of a
management unit is stocked in excess of a given
basal area level considered maximum to allow
acceptable forage production for allotment
management?" An economist interested in costs
might ask "How much of a management unit
needs to be treated, and to what intensity does
the treatment need to be applied, to bring the
tract to a prescribed stocking level?" A timber
manager might need data describing the extent
of merchantable sawtimber to a basal area level
considered the minimum for profitable
harvesting.

Decisionmaking at Different
Levels of Interest

If the "optimum" basal area level criterion
for sawtimber management potential is 50
square" feet per acre, we saw (fig. 4) that 43
percent of the management unit described above
would meet the treatment stocking objective.
This elementary type of statistic might be called
primary, or at the first level of interest.

If the sampling intensity is great enough,
other levels of interest can be exploited from
stocking equations. Our timber manager might
ask "If 43 percent of the management unit meets
the sawtimber treatment prescription, how
much of this latter area will be stocked with
residual trees of submerchantable size?" This is
an example of a secondary level of interest.
Source data from the proportion of the manage-
ment unit that meets the treatment prescription
can be subjected to regression or graphic
analysis. For our example, the proportion of the
management unit that meets the sawtimber
treatment prescription stocked with submer-
chantable ponderosa pine at various minimum
basal area levels is:

Basal area of

Cutover

submerchantable

area

ponderosa pine

stocked

(Sq. ft./acre)

(Percent)

20

82

40

69

60

59

80

50

100

42

The above information could provide the
basis for scheduling planting or determining site
preparation costs. For instance, if 60 square feet
of basal area per acre is judged satisfactory
stocking for advanced regeneration after
cutting, 59 percent of the proportion of the
management unit that meets the sawtimber
treatment prescription is already stocked, and
reproduction measures need be planned for 41
percent.

Setting Operating Priorities

The output of stocking equations — the
proportion of a management unit stocked by a
stand element to a specified criterion — can be
used with other information to set management
priorities. This can be illustrated by an example.

Ten Beaver Creek watersheds, similar to
watershed 12, were inventoried so that stocking
equations, and timber volume information,
could be developed. Let us rank these water-
sheds according to the desirability of harvesting
ponderosa pine sawtimber to achieve the "best
release" of pole-sized ponderosa pine trees and a
minimum release of a timber "weed" species,
Gambel oak (Quercus gambelii Nutt.). Direct
information, obtained from the stocking
equations, is the proportion of each watershed
stocked with pole-sized ponderosa pine, and the
proportion of each watershed stocked with
Gambel oak. Selected criteria are: (a) a minimum
sawtimber cut of 1,000 board feet per acre, (b) at
least 25 percent of the watershed stocked with
pole-sized ponderosa pine at a minimum basal
area level of 50 square feet per acre, and (c) no
more than 25 percent of the watershed stocked
with Gambel oak at a minimum basal area level
of 50 square feet per acre. This information is
arrayed in table 1.

The application here combines the area
information — the output of stocking equations
— with sawtimber volume estimates to
determine cutting priorities. Individual water-
sheds are eliminated from consideration when
they do not meet one or more criteria. Those
remaining are ranked on a sawtimber volume
basis. They could be ranked on any combination
of the above three criteria which could be shown
to maximize benefits or minimize costs.

Development of Distribution
and Density Functions

The relationships defined by stocking
equations are exceedance curves, which describe
the proportion of a stand stocked to minimum
basal area levels. Distribution functions, which

6

Table 1 . --Pr i or i t i es for harvesting large sawtimber

Watershed

Sawt imber

\i c\ 1 limp

Area stocked at minimum of
50 square feet of--

Feas i b i 1 i ty
priori ty

1 1

imits (a,b,c)— and
rankings (1 >2,3)

Pol e-s i zed

nonrlproca ninp

Gambe 1
oak

Bd. ft. /acre

— — — — Pprrpnt"

A

2,800

29

16

2

B

3,110

38

23

1

C

830

26

13

Insufficient

vol ume

D

1 ,660

22

26

1 nsuf f i cient

poles; too much oak

E

1 ,230

29

31

Too much oak

F

2,410

29

22

3

G

18

38

1 nsuf f i ci ent

poles; too much oak

H

16

16

1 nsuf f i c i ent

poles

1

1 , 0 jU

27

28

Too much oak

J

570

if6

13

1 nsuf f i c i ent

vol ume

— Limit

ng criteria: (a)
(b)

a minimum sawtimber cut of 1,000 board feet per acre;

at least 25 percent of the watershed stocked with pole-sized pon-

derosa pine at a minimum basal area level of 50 square feet per

acre;

(c) no more than 25 percent of the watershed stocked with Gambel oak at
a minimum level of 50 square feet per acre.

describe cumulative frequency, can be readily
developed from these relationships if desired.
With distribution functions, it would be possible
to derive density functions, which define the
probabilities of obtaining a small interval of
forest stocking considered prerequisite to
imposing a land treatment. Estimates of these
probabilities can be of value in decisionmaking
at the first level of interest .

Summary and Conclusions

1. Using point sampling techniques,
stocking conditions at a sample point
can be described in terms of whether or
not the point is stocked to a minimum
basal area level corresponding to a
particular BAF.

2. Mathematical relationships between pro-
portions of a forest stand stocked to mini-
mum basal area levels corresponding to
each BAF used in an inventory can be
defined through regression analyses. The

equations describing these regressions are
stocking equations; the dependent variable
is the proportion of a forest stand
stocked to a given minimum basal area
level, and the independent variable is
the minimum basal area level.

3. Stocking equations describing cutover and
virgin ponderosa pine stands in Arizona
were developed by means of a computer
program that approximates a sigmoidal
relationship. These equations define the
proportion of these two stands stocked to
minimum basal area levels up to 250 square
feet per acre.

4. Stocking equations can be used to help
evaluate land treatment potential, to
determine treatment feasibility on a
single management unit, and as a basis
for setting operating priorities on a
number of management units.

5. To apply this technique, the land manager
must have a multiple BAF inventory
made for the management unit in question,
then prepare stocking equations or graphs
similar to those described here.

Literature Cited

Anderson, T. C, A. A. Love, L. D. Wheeler, and
J. A. Williams.

1963. Soil management report for Long
Valley Ranger District, Coconino
National Forest. 97 p. U.S. For. Serv.,
Albuquerque, N. Mex.
Brown, Harry E.

1971. Evaluating watershed management
alternatives. Am. Soc. Civil Eng.,
J. Irrig. and Drain. Div.
97(IR1): 93-108.
Ffolliott, Peter F., and David P. Worley.

1965. An inventory system for multiple
use evaluations. U.S. For. Serv.
Res. Pap. RM-17, 15 p. Rocky Mt.
For. and Range Exp. Stn., Fort
Collins, Colo.
Grosenbaugh, L. R.

1965. Generalization and reparametrization
of some sigmoid and other nonlinear
functions. Biometrics 21: 708-714.

Jameson, Donald A.
1967. The relationship of tree overstory
and herbaceous understory vege-
tation. J. Range Manage. 20: 247-249.
Meyer, Walter H.
1961. Yield of even-aged stands of ponderosa
pine. U.S. Dep. Agric. Tech. Bull. 630,
59 p. (slightly revised).
Roberts, Edward G.
1964. A new insight to point sampling.
J. For. 62: 267-268.
Shiue, Cherng-Jiann.
1960. Systematic sampling with multiple
random starts. For. Sci. 6: 42-50.
Williams, John A., and Truman C. Ander-
son, Jr.

1967. Soil survey of Beaver Creek area,
Arizona. 75 p. U.S. Dep. Agric,
Wash., D. C.

Agriculture-CSU, Ft. Collins

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