United States 

Department of 

Agriculture 

Forest Service 

Northeastern Forest 

Experiment Station 

Research Paper 

N E-570 

Key Algorithms Used in GR02: 

A Computer Simulation Model 

for predicting Tree 

and Stand Growth 

Garrett A. Hughes 

Paul E. Sendak 
j The Authors 

Garrett A. Hughes received a B.S. degree in natural science from Michigan 

State University, and an M.S. degree in systems simulation from the Thayer 

School of Engineering at Dartmouth College. He has worked on a number 

of simulation projects related to the use of wood for energy, most recently 

for the State of Vermont, Burlington Electric Department, and the USDA 

Forest Service. He was a computer programmer with the George D. Aiken 

Sugar Maple Laboratory, Northeastern Forest Experiment Station, Burlington, Vermont, at the time the research reported here was started. 

Paul E. Sendak, forest economist and Project Leader, is engaged in , research on the economic impact of wood energy and the economics of 

maple syrup production at the Northeastern Station's Sugar Maple Laboratory at Burlington. He received a B.S. degree in forestry at Rutgers University, and an M.S. degree in forestry and Ph.D. degree in forestry economics 

at the University of Massachusetts. He joined the USDA Forest Service in 

1969. 

Manuscript received for publication 27 December 1984 

Abstract 

The Vermont Woodland Resource Analysis Program (VWRAP) is an 

interactive computer program designed to help forest-land owners make 

decisions. GR02 is a simulation model written as a subprogram in VWRAP 

for predicting individual tree and stand growth over time. The model can be 

characterized as an individual tree, distance-independent model. It performs five major functions during each run: (1) updates diameter at breast 

height, (2) updates total height, (3) estimates mortality, (4) determines 

regeneration, and (5) updates crown class. The key algorithms used in 

GR02 are presented. 
Introduction 

An inventory of standing timber is one of the main 

requirements in the preparation of a forest management plan. However, an inventory is static, and it is 

only valid for the time when the data were collected. 

One method of making the inventory dynamic is 

through simulation. By simulating changes that take 

place in the growing forest, an inventory taken at one 

time can be projected to some future time. 

In Vermont, 90 percent of the commercial forest 

land is privately owned (Kingsley 1977). Excluding 

forest industry, corporate ownership, and forest land 

owned by private organizations, there are 74,000 

individual owners of 3 million acres of commercial 

forest land in Vermont. A system that integrates a 

static concept of forest inventory with the dynamics of 

growth simulation can improve decisions made by 

forest-land owners and managers of this land. 

The Vermont Woodland Resource Analysis Program (VWRAP) is designed to help forest-land owners 

make decisions; VWRAP consists of three computer 

programs: DATA2, YIELD, and GR02. Data from a forest inventory collected in a prescribed format are 

entered into a computerized data base via DATA2, an 

interactive program. The user can then produce a summary inventory of the woodlot on a stand-by-stand 

basis with the interactive program YIELD. 

Long-term projections of grom for trees in the 

data base can be obtained with GR02, a computer 

simulation model for predicting individual tree and 

stand growth over time. The model is an individual 

tree, distance-independent model with both continuous 

and discrete variables (Ek and Dudek 1980). It can project the growth of stands composed of mixtures of 

species in all-aged or even-aged stands. The model, 

written in FORTRAN-20, version 5, is designed to run 

on a DEC-System 20 computer using the TOPS-20 

operating system. 

In this paper we present the key algorithms used 

to model the growth of individual sample trees and 

sample plots from which stand growth is inferred. The 

"Vermont Woodland Resource Analysis Program 

(VWRAP) User's Manual" describes how to use the 

model.' 

Model Philosophy 

Trimble (1969) reported that crown class is the 

most important predictor of the diameter growth of 

trees. We tested and confirmed Trimble's conclusion 

using radial increment measurements from core samples in Vermont. Given a number of diameter-growth 

predictor variables to choose from-d.b.h., crown 

length, crown width, basal area, crown class, and vigor 

class-for a given species on the same site, crown 

class was the best overall predictor of current diameter 

growth at breast height (Trimble 1969). Trimble's result 

suggested a method that was both simple and reliable. 

The model relies heavily on crown class, dominant, codominant, intermediate, and overtopped (Trimble 1969), because of the statistical correlation 

between crown class and diameter growth. The 

detailed model description that follows shows that the 

computational algorithms that use crown class for 

growth prediction are relatively simple and quick to 

execute. 

Diameter growth is projected on the basis of a 

tree's crown class. The total height of each sample 

tree is updated during a simulation run; new relative 

heights are computed, and Monte-Carlo techniques are 

used to determine a tree's crown class based on its 

current relative height. 

For simplicity in introducing the model, the 

preceding discussion has ignored several other important factors in tree growth. Among these are species, 

site index, stocking density expressed by basal area 

per acre, species-specific response to release, and 

damage as a result of injury or disease. These factors 

are treated in the sections that follow. 

A Dynamic Hypothesis 

To use crown class as a growth predictor, it is 

necessary to know the crown class of a tree at any 

stage in its development. The growth simulator must 

be able to determine a tree's crown class at any point 

in time. This problem was solved empirically, using 

measurements taken from sample plots scattered 

throughout northern and central Vermont. Figure 1 

illustrates the solution and represents the dynamic 

hypothesis which is the key concept in the simulation 

model. 

A copy of the manual can be obtained by writing: Extension Forester, School of Natural Resources, University of 

Vermont, Burlington, Vermont 05405. 

Figure 1 shows the proportion of trees in a 

particular crown class as a function of relative height. 
1 .o \ 

\ 

\ Dominant 

0.9 - \ -- Codominant \ 

'\ --- Intermediate 

\ ------ 0.8 - \ Overtopped 

\ 

\ 

\ 

> 0.7 - ', 

Figure 1.-Relative frequency of crown class as a function of relative tree 

height or height of tree expressed as a proportion of canopy height. 

The proportions sum vertically to 1 at each relative 

height. As a tree's relative height increases, it is more 

likely to be in one of the higher crown classes, thus 

increasing its annual radial-growth rate. As a tree's 

relative height decreases, because it is overtopped by 

surrounding trees, it is more likely to be in a lower 

crown class, thus decreasing its annual radial-growth 

rate. 

An individual tree's relative height depends on the 

height of surrounding trees on the plot. Relative height 

of the ith tree on a sample plot is defined by the 

expression: 

where 

RHi = relative height of the ith tree, 

H~ = total height of the ith tree, 

and 

- 

H = canopy height. 

2 

The canopy height, H, is the weighted average height 

of the trees on the plot, where each tree's total height 

has been weighted by the tree's basal area. For 

variable-radius plots, it follows that the canopy height 

is the simple average height of the sampled trees. 

The largest diameter trees on a plot make up the 

canopy, whereas the smaller diameter trees shape the 

understory. This agrees with the observation that one 

can remove a large number of smaller diameter (understory) trees from an area with little effect on the height 

of the canopy. The relative heights of the remaining 

trees are changed little by removing the smaller trees. 

With this model, growth rates of the remaining trees 

also change little. However, removing the larger trees 

from the same area reduces the canopy height, 

changes the entire relative height distribution, and 

thrusts the remaining trees into higher crown classes 

with a resulting increase in their overall growth rate. 
Key Algorithms in the Growth Model 

initializing GR02 

Tree data are entered into GR02 by plot. In GR02, 

each sample plot is expanded to 1 acre, regardless of 

whether the information was obtained from variableradius or fixed-area plots, before the simulation begins. 

Each of these I-acre plots is treated independently as 

if there were n independent forests all growing at the 

same time within the model. 

This process associates a frequency per acre with 

each sample tree on a plot. This frequency or expansion factor is used to compute the per-acre contribution of each sample tree to a given variable. For 

example, multiplying a tree's total biomass by its frequency per acre gives its contribution to the total biomass per acre for that plot. 

Stand information tables, on the other hand, are 

the result of averaging all of the information contributed by the independent sample plots. Growth characteristics depend only on information associated with a 

single plot; stand table information depends on data 

supplied from all the plots entered into the simulation. 

By not aggregating plot information during the 

growth simulation, the heterogeneity of the forest is 

maintained. Different parts of the forest will grow at 

different rates depending on site characteristics and 

tree species composition. 

Basal Area 

Tree basal area is the basal area (in square feet) 

associated with a sampled tree. Tree basal area per 

acre is defined as the product of tree basal area of a 

sample tree and its corresponding frequency per acre. 

For example, when a 10-factor prism is used to sample 

trees in the inventory process, a single 10-inchdiameter tree on an expanded sample plot represents 

18.3 simulator trees per acre. During the model initialization process, a frequency of 18.3 trees per acre is 

associated with this sample tree. The tree basal area 

per acre of this sample tree equals the product of the 

tree basal area (0.5454 ft2) and the sample tree frequency (18.3 trees per acre), or 10 ft2 per acre. Total 

basal area per acre is defined as the sum of all the tree 

basal areas per acre on the same sample plot. 

The GR02 model performs five major functions 

during each run: (1) updates diameter at breast height, 

(2) updates total height, (3) estimates mortality, (4) 

determines regeneration, and (5) updates crown class. 

Each function is treated in detail in the following sections. Together, they provide the mechanisms for 

modeling the dynamics of growth of a forest. 

Updating Diameter at Breast Height 

The diameter at breast height outside bark is 

updated once each growing season during a simulation. First, radial increment inside the bark is determined for each sample tree; radial increment is 

converted to diameter increment outside the bark; and 

diameter increment is added to the current diameter 

outside bark to give updated breast height diameter. 

Estimating the annual radial increment. On the 

basis of an analysis of 432 core samples taken 

throughout the State of Vermont, the annual radial 

increment (inside bark), RINC, was determined to be 

primarily a function of species, crown class, site index, 

and stand density. 

Within a given crown class (dominant, codominant, intermediate, and overtopped), the minimum 

radial increment for trees in stands with a total basal 

area per acre of 100 ft2 and greater is determined from 

the expression: 

MNRINC = Xo + XI (SITE) (2) 

where 

MNRINC = minimum radial increment (mm) 

(100 ft2 BAlA and greater), 

SITE = site index at breast height age 50, 

and 

Xo, XI = species dependent coefficients. 

Similarly, the maximum radial increment for 40 ft2 

and lower basal area is determined from the 

expression: 

MXRINC = Xo + Xi (SITE) 

where 

MXRINC = maximum radial increment (mm) (40 

ft2 BAlA and lower). 
Within a given crown class and between 40 and 

100 ftZ of basal area, linear interpolation gives: 

MXRINC - MNRINC 

RlNC = MNRINC + - 40 

LBAPA) 

(4) 

where 

RlNC = radial increment (40 ft2<BAIA <I00 ft2) 

and 

LBAPA = total basal area per acre. 

Coefficients for expressions (2) and (3) were estimated for annual radial increment (in millimeters) for 

the following species: red spruce, white pine, 

hemlock, red maple, sugar maple, yellow birch, white 

birch, American beech, white ash, and northern red oak 

(Appendix, Table 6). Two additional sets of coefficients 

were developed for all hardwood species and for all 

softwood species. These are used to determine radial 

increment for species not included here. 

If more appropriate coefficients are available from 

another source for a species we did not include, users 

can add them to the program or can substitute one of 

the available sets. The simulation program, and 

VWRAP in general, were designed to easily accommodate user modifications. 

Trees are classified during the field inventory by 

condition and damage class. Trees with live, intact 

tops showing no signs of damage are assigned a radial 

increment as calculated previously. All other trees 

suffer an arbitrary degradation in radial increment by a 

factor between 0 and 1 selected at random from a 

rectangular distribution by the program. In other words, 

because no data were collected to correlate growth 

with damage, the expected value for the growth rate of 

damaged trees is assumed to be half the rate of 

undamaged trees. 

To obtain the diameter increment outside bark, 

radial increment is first doubled, then converted to 

diameter increment outside bark by dividing by the 

species-specific ratio of diameter inside bark to diameter outside bark. The species-specific ratios were estimated by regression analysis of the trees in the 

growth-rate data base (Table 1). The values for an additional 12 species were obtained from Martin (1981). 

Other species are assigned a value of 0.926 for this 

ratio. 

When trees change crown class, the model gradually changes diameter increment (Forrester 1961). 

Diameter increments are weighted by species-specific 

response times which have been developed from the 

shade-tolerance ratings assigned to various species 

(Spurr and Barnes 1980). Using these weighting factors 

causes shade-tolerant trees to take three growing seasons to respond fully to a change in crown class, 

whereas shade-intolerant species may take as long as 15 

years. 

Table 1.-Average DlBlDOB ratios, by species, for 

growth-rate trees 

Species 

American beech 

Northern red oak 

Red maple 

Sugar maple 

White ash 

White birch 

Yellow birch 

Eastern hemlock 

Eastern white pine 

Red spruce 

DlBlDOB ratio 

Updating Total Height 

The total height of each sample tree is updated 

once each growing season during a simulation run; 

GR02 uses information obtained from site-index 

curves to update height. Site-index curves used in 

GR02 were obtained from Hampf (1965) for red spruce, 

eastern hemlock, and eastern white pine; from 

Carmean (1978) for red maple, American beech, and 

northern red oak; and from Curtis and Post (1962) for 

sugar maple, yellow birch, white birch, and white ash. 

lnformation in site-in'dex curves. Site-index curves 

are the family of curves showing the total height for a 

given species plotted as a function of age at breast 

height for different site-index values. Within a foot or 

two, most of the published curves can be expressed 

as: 

HGHT = 

where 

HGHT = 

TMXH = 

T = 

and 

THTC = 

TMXH t- (4.5 - TMXH)e-TITHTC (5) 

total height (feet), 

tree maximum height (feet), 

breast height age (years), 

tree height time constant (years). 
Breast height age is the independent variable; the 

maximum height (TMXH) and height time constant 

(THTC) are parameters of the particular curve chosen. 

When T equals zero in expression (5), the height is 4.5 

feet; when T is very large, the total height is approximately equal to the maximum height. 

Height increment (HINC) can be written as: 

HINC = [- ] (TMXH - HGHT) THTC (6) 

Equation (6) is the approximate solution to equation (5), 

which fits the site-index curves. It is a powerful tool for 

simulation because it gives the rate of height growth 

as a function of the height and not the age of the tree. 

The two constants, TMXH and THTC, are obtained 

for any site-index curve by fitting equation (5) exactly 

to two points on the curve. The values of TMXH and 

THTC were obtained by using an iterative approximation technique. The reader is cautioned against a 

strictly biological interpretation of the two constants 

because the shape of some of the curves implies 

values that we would not expect to observe in the field. 

In GR02, site-index values are grouped into 10- 

foot classes (Table 2). Values of TMXH and THTC for 

10 species and seven site-index classes within species 

are contained in GR02(Appendix, Table 7). Currently, 

other hardwoods use the values for yellow birch, and 

other softwoods use the values for white pine. Values 

for other species can be added, or existing values 

changed. 

Annual height-increment modifier. Understory 

trees grow taller more slowly than overstory trees. The 

height-growth modifier quantifies this observed change 

in rate of height growth. If the sample tree belongs to 

the intermediate or overtopped crown classes, the 

computed height increment is multiplied by the tree 

height-growth modifier, THGM. 

We assumed that the maximum height constant, 

TMXH, is not affected for understory trees, but that the 

height time constant, THTC, becomes larger. If THTCl 

is the time constant for dominant and codominant 

trees of the species and site under consideration, and 

THTC2 is the time constant for an understory tree, the 

height-growth modifier is defined as the ratio: 

Table 2. Site-index classes used in GR02 

Number Site-index values Site-index class 

1 00-44 40 

2 45-54 50 

3 55-64 60 

4 65- 74 70 

5 75-84 80 

6 85-94 90 

7 95 + 100 

Height-growth modifiers were calculated from the 

calibration data for each of 10 species, other hardwoods, other softwoods, and for the intermediate and 

overtopped crown classes. Substituting for THTC1 and 

THTC2 in equation (7) and simplifying yields: 

In [(HGHT2 - TMXH)1(4.5 - TMXH)] THGM = 

In [(HGHTI - TMXH)1(4.5 - TMXH)] (8) 

Potential height, HGHT1, was read from the site-index 

curve; actual height, HGHT2, was observed from the 

calibration data; and TMXH was estimated from the 

same site-index curve. Averages for each species and 

understory crown class were calculated and are stored 

in GR02. 

Mortality 

The mortality segment of GR02 was adapted from 

Buchman (1979). The estimated annual probability of a 

tree dying is given by the expression: 

where 

MR = 

DlNC = 

and 

A, B, C, D = 

mortality rate, 

exponentially weighted diameter 

increment outside bark (incheslyear), 

species specific parameters (AppenTHGM = THTClITHTC2; THTC2 >THTC1 (7) dix, Table 8). 
The mortality rate depends on the rate of diameter 

growth. At a high rate of diameter growth, the mortality 

is approximately equal to the constant, D. Mortality 

rate approaches its maximum value as diameter growth 

drops to zero. Between zero diameter growth and a 

high rate of diameter growth, mortality rate decreases 

exponentially. 

In GR02, trees in the understory will be growing 

more slowly than those in the overstory, and have 

higher mortality rates. Fast-growing dominant trees will 

have the highest survival rate. Undamaged trees.generally will have a higher diameter increment than 

damaged trees; thus, the damaged trees will die at a 

higher rate. 

Coefficients provided with Buchman's expressions 

(1979) were adjusted so that the more shade-tolerant 

species had a lower mortality rate at lower levels of 

diameter increment. Shade-intolerant species had a 

higher mortality rate at the lower levels of diameter 

increment. Species intermediate in shade tolerance 

were not adjusted. 

Once the mortality rate is determined for a given 

tree, a new frequency is computed for that tree by 

multiplying the current frequency by 1 minus the 

mortality rate. A new basal area per acre also is computed by multiplying the tree basal area by the newly 

computed frequency. 

We assumed that the higher the site index for a 

given plot, the higher the maximum basal area that can 

be supported on that plot. It also is assumed that softwood plots can support higher basal areas than hardwood plots; GR02 keeps the total basal area per acre 

from exceeding some maximum value for each sample 

plot. 

Given these conditions, the maximum total basal 

area per acre that can be supported on a 1-acre sample 

plot is assumed to follow the relation: 

LMBA = MBAH + RSBA(MBAS - MBAH) (10) 

where 

LMBA = maximum total basal area (ft2 per acre), 

MBAH = maximum basal area for hardwoods (ft2 

per acre), 

MBAS = maximum basal area for softwoods (ft2 

per acre), 

For hardwoods, the maximum basal area per acre 

is given by the expression: 

MBAH = 2.5 WAS1 (1 1) 

where 

WASl = weight-averaged site index (ft at breast 

height age (Bha) = 50). 

For softwoods, the maximum basal area per acre 

is given by the expression: 

MBAS = 3.375 WASl 

The weight-averaged site index for each sample 

plot is determined from the expression: 

where 

WASl = weight-averaged site index (ft at Bha 

= 50), 

SITEi = species site index for the ith sample 

tree (ft at Bha = 50), 

BAi = basal area per acre contributed by the 

ith sample tree (tree basal area x frequency), 

and 

N = number of sample trees (not the frequency) on the sample plot. 

Substituting for MBAH and MBAS in expression 

(10) yields: 

LMBA = 2.5 WAS1 (1 - RSBA) + 3.375 RSBA 

WAS1 (1 4) 

Expression (14) is used in GR02 to determine the maximum allowable total local basal area per acre for a 

given sample plot. 

and 

RSBA = ratio of softwood total basal area to 

hardwood total basal area per acre. 
After the number of simulator trees is reduced 

through mortality, the total basal area per acre is computed and compared with the maximum allowable for 

each plot. If the total exceeds the maximum, the plot 

reenters the mortality subprogram; otherwise, no more 

mortality occurs on the plot for the current growing 

season. 

Determining Regeneration 

New trees are brought into the model in proportion to the basal area of existing species. This simple 

model of regeneration ignores the many complexities 

observed in natural regeneration; however, it does 

provide for ingrowth on the sample plots. For the 

length of simulation that we envisioned, the species 

composition and timing of regeneration are of little 

consequ nce. To focus on the simulation of regeneration, f more complex model would be needed. 

Determining the number of new trees to enter. 

Only species currently existing on a sample plot are 

regenerated on that plot. Trees are entered into the 

I-inch d.b.h. class at a fixed diameter of 0.5 inch at 

breast height outside bark, a height of 8.5 feet, and an 

overtopped crown class. 

To determine how many trees to enter on a sample 

plot, GR02 examines the total basal area per acre in 

the 1-inch diameter class (0.50 to 1.49 inches) at the 

end of each growing season. GR02 maintains between 

1.84 and 5.0 ft2 basal area per acre in the I-inch 

diameter class. 

When regeneration occurs, a tree is created for 

each species existing on the plot. The basal area per 

acre associated with each new tree is computed from 

the expression: 

BAi = PROBA, (MXBA1 - TOTBAI) (1 5) 

where 

BAi = basal area per acre associated with 

the ith new tree (ft2 per acre), 

PROBA, = proportion of total basal area per acre 

contributed by the jih species on this 

plot, 

MXBAI = maximum basal area assigned to 1- 

inch d.b.h: class (ft2 per acre), 

and 

TOTBAI = total basal area per acre in 1-inch 

class (ft2 per acre). 

The frequency associated with each new tree is 

given by the expression: 

where 

FREQi = frequency associated with ith new tree 

(trees per acre), 

and 

TBAi = tree basal area of ith new tree (ft2 per 

acre). 

The GR02 model allows regeneration only when 

the basal area of the sapling crop in the 1-inch 

diameter class fails below some arbitrary value (1.84 

ft2). If GR02 allowed regeneration to occur annually, 

the number of trees would increase rapidly with little 

observable impact on the growth characteristics of the 

larger trees, but with a marked impact on the model's 

run-time performance. 

In practice, the number of new trees entering a 

simulation usually is small (though the associated frequency may seem large). The reason for this is that 

mortality of trees in the 1-inch d.b.h. class is balanced 

by basal-area growth of survivors, which keeps the 

basal area above the level for entering new trees. Also, 

trees in this diameter class generally are slow to grow 

out of it. 

Updating Crown Class 

At the end of each simulated growing season, a 

new crown class is selected for each sample tree on 

each plot. The selected crown class may be the same 

or different from the prior one. The probability of a 

sample tree being assigned a given crown class 

depends on its relative height (Fig. 1). 

At the end of each growing season, GR02 computes a new canopy height for each sample plot. Once 

the canopy height is known for a plot, the relative 

height of each tree is determined from the ratio of 

sample tree height to canopy height. 

Sample trees are assigned to one of 16 relative 

height classes ranging from 0 to 1.5 in steps of 0.1. A 

discrete probability distribution based on Figure 1 is 

included in GR02 for each of the relative height 

classes. The distribution gives the probability that a 

tree will be assigned to a given crown class. 

A random number between 0 and 1 is selected by 

GR02. This number is matched to the cumulative 
Calibration Data 

distribution function for the relative height class, and 

an appropriate crown class is selected for the sample 

tree under consideration. The process of selecting 

crown classes continues until all sample trees on all 

sample plots have been assigned a crown class. 

The possibility of changing crown classes each 

year is not as unrealistic as it may sound at first. Trees 

that are at the extremes of relative height, either high 

or low, have a high probability in GR02 of remaining in 

the same crown class from year to year. When changes 

do occur, it is most often to an adjacent crown class. 

Rarely does a tree shift two classes in one year. However, because of the relative height probability distribution, this outcome is possible. Five trees on one plot 

from the 1983 Vermont forest inventory were monitored 

for changes in crown class for a simulation period of 

51 years (Table 3). 

The stochastic treatment of crown Class serves 

two functions. First, it models the random variation in 

radial increment. If radial growth of a particular tree is 

defined as the sum of a constant and a random term, 

we have said in GR02 that radial increment changes 

"as if" the crown class were changing. 

The second function deals with changes in canopy 

height and relative heights of sample trees. These 

changes may be the result of height growth, thinning, 

or harvesting. But they alert the model that a tree's 

competitive position and, therefore, its radial growth 

may change. 

The algorithms in GR02 illustrate how the model 

simulates change in a forest stand. Although presented independently of DATA2 and YIELD, the three 

computer programs complement each other in VWRAP 

by allowing input of plot data, processing of that data, 

and output of stand summary tables. However, GR02 

can be used separately as a subprogram if desired. 

In addition to generating the logic and FORTRAN 

code, the development of GR02 also required the 

specification of the functional relationships that 

described the dynamics of forest stands, and the 

empirical estimation of the parameters of those functions. The calibration of the model was accomplished 

with a set of data from 530 trees. There were 432 

growth-rate trees (Table 4) and 309 site-index trees, 

with 21 1 trees used in both capacities. These data 

were collected by the Service Foresters in the Vermont 

Department of Forests, Parks, and Recreation. Because 

of the relatively small sample size, a probability sample 

was not attempted. Instead, a matrix of species (10) by 

crown class (4) was constructed and sample trees were 

selected from around Vermont to fill each cell. The 

trees were selected arbitrarily from forest stands with a 

basal area of 100 ft2 per acre or more. Only live, nondiseased trees were selected. The key data collected for 

each tree included: species; crown class; increment 

core at breast height through the pith; total height; 

d.b.h.; and site index. 

Table 3.-Results of changes in crown class for five trees for a 51-year 

simulation using GR02 

Total number Total number 

Tree Initial Final of crown class of crown class 

number Species changes in changes exceeding class class 51 years 1 classa 

1 Yellow birch 4 2 22 4 

2 Eastern hemlock 2 2 27 7 

3 Sugar maple 2 1 7 0 

4 Sugar maple 3 1 24 2 

5 Sugar maple 2 1 5 0 

a NO change in excess of two classes occurred. 
Table 4.-Number of growth-rate trees, by crown class 

Crown class 

Species . ,,, 

Dominant Codominant Intermediate Overtopped Total 

American 

beech 13 12 12 11 48 

Northern 

red oak 14 16 11 12 53 

Red maple 13 12 12- 10 47 

Sugar maple 16 15 13 15 59 

White ash 15 13 10 9 47 

White birch 10 9 9 7 35 

Yellow birch 8 13 8 9 38 

Eastern 

hemlock 9 10 8 10 37 

Eastern 

white pine 14 10 9 

Red spruce 6 7 6 7 26 

Total 118 117 98 99 432 

I 

~ The increment cores for growth-rate and site-index 

trees were analyzed in the laboratory. Radial increment 

was measured to the nearest 0.1 mm for the most 

I recent 12 growth rings. The age at breast height, from 

i a ring count, and the distance from the cambium to 

pith also were recorded. Each core was analyzed twice ! to reduce measurement errors. 

The data were analyzed statistically to obtain the 

annual radial-increment coefficients in the Appendix 

(Table 6). The growth-rate trees were used to obtain the 

coefficients on sites exceeding 100 ft2 basal area per I 

acre. The site-index trees were used to obtain the 

radial-growth coefficients on sites with less than 40 ft2 

basal area per acre. These trees were either in the 

dominant or codominant crown class, free from 

disease, damage, and suppression, and between 30 

and 100 years at breast height. In other words, they 

represented the best expression of growth potential on 

the site for the species. 
Sample Output From GR02 

The GR02 model has been tested for soundness 

of its results; that is, for the stands in which it has 

I been used to simulate change, the results seem 

reasonable. A systematic program of testing using 

i 

! actual stand growth data is planned. 

i The data summaries available from GR02 include 

a Stand Table, Basal Area Table, Biomass Table (dry 

tons per acre), Biomass Value Table, Net Biomass 

Table (net of sawlogs), Net Biomass Value Table, Sawi timber Volume Table, Sawtimber Value Table, Saw- 4 timber Stand Table, Sawtimber Basal Area Table, and 

4 

D.b.h.lStems per acre Histogram. All or any combination of tables can be printed. Biomass weight is estimated by equations developed by Monteith (1979). 

As an example, 11 plots were taken in a typical 

mixed stand in the Sugar Maple-Beech-Yellow Birch 

forest type in northern Vermont. Table 5 is the stand 

table from GR02 at the time of data collection, 

November, 1982. The stand is pole- to small sawtimbersized, with 97.3 ft2 basal area per acre. It has not been 

managed actively, though merchantable timber has 

been cut in the past. 

Table 5.-Beginning stand table produced by GR02 for a northern hardwood 

stand in Vermont 

D.6.h. Sugar Yellow Black American Red Red Total class maple birch cherry beech maple spruce 

Inches - - - - - - - - - - - - - - - No, stemslacre - - - - - - - - - - - - - - - 

Total 150.3 72.1 12.4 35.1 37.2 14.2 321.3 
Using GR02, the stand growth was simulated for a sawtimber volume by species; Figure 4 shows the 

25-year period assuming no silvicultural practices or change in the diameter distribution of the stand from I harvesting. Figures 2 through 4 show some of the data 1982 to 2007. In addition, total biomass of the stand 

summaries that can be made from GR02 output. increased from 63.1 dry tons per acre in 1982 to 89.1 

Figure 2 shows how species composition changed tons in 2007. 

over the 25-year period. Figure 3 shows the trend in 

U Sugar maple - [1--------0 Yellow birch 

Black cherry 

American beech 

Red maple 

0--------0 Red spruce 

I I I I 

1982 1987 1992 1997 2002 2007 

YEAR 

Figure 2.-Simulated changes in basal area in a northern hardwood stand 

in Vermont for a 25-year period. 
b-A Sugar maple 

Yellow birch 

Black cherry 

American beech 

0-0 Red maple 

0---------0 Red spruce 

I I I I I I 

1982 1987 1992 1997 2002 2007 

YEAR 

Figure 3.-Simulated changes in sawtimber volume in a northern hardwood 

stand in Vermont for a 25-year period. 
DIAMETER AT BREAST HEIGHT (Inches1 

Figure 4.-Diameter (d.b.h.) distribution for a northern hardwood stand in 

Vermont for 1982 and 2007. 
Conclusion 

We believe that in its intended application, as a 

growth simulator in VWRAP, GR02 is useful for the 

kinds of forest management decisions made by landowners and managers in Vermont. It can give landowners and managers a reasonable view of the future 

consequences of their potential management 

decisions. Together with an economic analysis, GR02 

should improve the decision-making process 

and could lead to more acreage of forest land under 

management. 

Literature Cited 

Buchman, R. G. Mortality functions in a generalized 

forest growth projection system. Gen. Tech. Rep. 

NC-49. St. Paul, MN: U.S. Department of Agriculture, 

Forest Service, North Central Forest Experiment 

Station; 1979. 47 p. 

Carmean, W. H. Site index curves for northern hard. 

woods in Northern Wisconsin and Upper Michigan. 

Res. Pap. NC-160. St. Paul, MN: U.S. Department of 

Agriculture, Forest Service, North Central Forest 

Experiment Station; 1978. 16 p. 

Curtis, C. 0.; Post, B. W. Site index curves for evenaged northern hardwoods in the Green Mountains of 

Vermont. Bull. 629. Burlington, VT: University of 

Vermont, Agricultural Experiment Station; 1962. 

Ek, Alan R.; Dudek, Albert. Development of individual 

tree based stand growth simulators: progress and 

applications. Staff Pap. Ser. No. 20. St. Paul, MN: 

University of Minnesota, Department of Forest 

Resources; 1980. 25 p. [Paper presented at Workshop 

on Forest Succession, sponsored by the Oak Ridge 

National Laboratory; 1980 June 8-14; Mountain Lake, 

VA]. 

Forrester, Jay W. Smoothing of information. In: Industrial dynamics. Cambridge, MA: The M.I.T. Press; 

1961 : 406-41 1. 

Hampf, F. E. Site index curves for some forest species 

in the Eastern United States, rev. Upper Darby, PA: 

U.S. Department of Agriculture, Forest Service, 

Eastern Region (7); 1965. 43 p. 

Kingsley, Neal P. The forest resources of Vermont. 

Resour. Bull. NE-46. Upper Darby, PA: U.S. Department of Agriculture, Forest Service, Northeastern 

Forest Experiment Station; 1977. 58 p. 

Martin, A. Jeff. Taper and volume equations for 

selected Appalachian hardwood species. Res. Pap. 

NE-490. Broomall, PA: U.S. Department of Agriculture, Forest Service, Northeastern Forest Experiment Station; 1981. 22 p. 

Monteith, D. B. Whole tree weight tables for New York. 

AFRl Res. Rep. No. 40. Syracuse, NY: State University of New York, Applied Forestry Research 

Institute, College of Environmental Science and 

Forestry; 1979. 64 p. 

Spurr, S. H.; Barnes, B. V. Competition and survival. 

In: Forest ecology. 3d ed. New York: John Wiley and 

Sons; 1980: 363-398. 

Trimble, G. R. Diameter growth of individual hardwood 

trees. Res. Pap. NE-145. Upper Darby, PA: U.S. 

Department of Agriculture, Forest Service, Northeastern Forest Experiment Station; 1969. 25 p. 

Acknowledgment 

This study was the result of cooperation between 

the Northeastern Forest Experiment Station; Burlington 

(VT) Electric Department; Vermont Department of 

Forests, Parks, and Recreation; and Extension Service, 

University of Vermont. The contributions of the following people to the project are gratefully acknowledged: 

William Kropelin, Brian Stone, Thomas McEvoy, Harold 

Remington, and the Vermont Service Foresters. 
Appendix 

Table 6.-Annual radial-increment coefficients (mm) used in GR02, by total basal area per acre and 

crown class 

Total basal arealacre 

Species Crown class 

Dominant Codominant Intermediate Overtopped 

xo XI xo x1 xo X, xo x1 xo 

American 

beech 1.5977 

Northern 

red oak 1.9500 

Red maple 1.4492 

Sugar 

maple - 0.1525 

White ash - 0.3110 

White 

birch 1.1150 

Yellow 

birch 1.501 2 

Eastern 

hemlock 1.3611 

Eastern 

white pine 2.9271 

Red spruce 1.3983 

Other 

hardwoods 0.3545 

Other 

softwoods 2.1248 
--L 

0 Table 7.-Values of time height constant (THTC) and maximum height (TMXH), by species and site index 

Site-index class 

Species 40 50 60 70 80 90 100 

American 

beech 

Northern 

red oak 

Red 

maple 

Sugar 

maple 

White 

ash 

White 

birch 

Yellow 

birch 

Eastern 

hemlock 

Eastern 

white pine 

Red 

spruce 

THTC TMXH 

796.88 588.16 

205.08 168.58 

1 15.86 105.63 

53.71 63.11 

38.09 53.06 

32.50 50.94 

45.12 57.50 

33.11 50.07 

141.60 123.82 

33.1 1 50.07 

THTC TMXH 

161.13 175.07 

78.71 101.27 

91.60 112.66 

53.76 79.65 

38.04 66.71 

33.11 64.19 

45.17 72.46 

50.24 76.69 

98.83 119.10 

50.24 76.69 

TMXH 

132.96 

99.87 

130.54 

96.12 

80.38 

77.47 

87.36 

90.01 

124.01 

90.01 

THTC TMXH 

63.09 124.17 

43.12 99.92 

71.97 135.29 

53.76 112.68 

38.04 94.06 

33.84 90.71 

45.12 102.28 

49.31 107.30 

70.61 133.57 

46.09 103.44 

TMXH 

120.64 

102.01 

144.56 

129.20 

107.73 

103.98 

11 7.22 

129.00 

157.69 

133.75 

THTC TMXH 

39.75 123.96 

19.73 97.36 

57.23 151.27 

53.76 145.72 

38.04 121.40 

34.25 117.24 

45.12 132.14 

60.45 156.45 

60.45 156.45 

60.45 156.45 

THTC TMXH 

60.35 174.04 

43.36 144.05 

52.25 159.53 

56.40 166.93 

39.65 137.77 

37.99 136.63 

48.95 153.73 

52.25 159.53 

47.56 151.30 

47.59 151.30 
Table 8.-Tree mortality coefficients used in GR02 

Coefficient 

Species A B C D 

American basswood 

American elm 

Bigtooth aspen 

Black ash 

Hickory 

Paper birch 

Quaking aspen 

Red maple 

Red oak 

Sugar maple 

White ash 

White oak 

Yellow birch 

Balsam fir 

Black spruce 

Eastern hemlock 

Eastern white pine 

Jack pine 

Northern 

white cedar 

Red pine 

Tamarack 

White spruce 

Other softwoods 

Other hardwoods 
Hughes, Garrett A.; Sendak, Paul E. Key algorithms used in 

GR02: A computer simulation model for predicting tree and 

stand growth. Res. Pap. NE-570. Broomall, PA: US. 

Department of Agriculture, Forest Service, Northeastern 

Forest Experiment Station, 1985. 17 p. 

GR02 is an individual tree, distance-independent simulation 

model for predicting tree and stand growth over time. It performs five major functions during each run: (1) updates 

diameter at breast height, (2) updates total height, (3) 

estimates mortality, (4) determines regeneration, and (5) 

updates crown class. 

ODC 547:564(743) 

Keywords: Forest growth; mortality; regeneration; volume 

estimation (stands); volume increment (stands); site index; 

crown class 

I *.U.S. GOVERNMENT PRINTING OFFICEr1985-505-028:20018 
Headquarters of the Northeastern Forest Experiment Station are in 

Broomall, Pa. Field laboratories are maintained at: 

Amherst, Massachusetts, in cooperation with the University of 

Massachusetts. 

Berea, Kentucky, in cooperation with Berea College. 

@ Burlington, Vermont, in cooperation with the University of 

Vermont. 

Delaware, Ohio. 

Durham, New Hampshire, in cooperation with the University of 

New Hampshire. 

Hamden, Connecticut, in cooperation with Yale University. 

@ Morgantown, West Virginia, in cooperation with West Virginia 

University, Morgantown. 

@ Orono, Maine, in cooperation with the University of Maine, 

Orono. 

@ Parsons, West Virginia. 

Princeton, West Virginia. 

@ Syracuse, New York, in cooperation with the State University of 

New York College of Environmental Sciences and Forestry at 

Syracuse University, Syracuse. 

University Park, Pennsylvania, in cooperation with the 

Pennsylvania State University. 

@ Warren, Pennsylvania.