PROGNOSIS MODEL FOR

STAND DEVELOPMENT

Albert R. Stage
THE AUTHOR 

Albert R. Stage is Principal Mensurationist at the Forestry Sciences Laboratory, 

Moscow, Idaho. His research has included studies of site evaluation, 

 stand stocking measures, and sampling methods for growth and yield estimation. 
USDA Forest Service 

Research Paper INT-137 

June 1973

PROGNOSIS MODEL FOR STAND 

DEVELOPMENT 

Albert R. Stage 

INTERMOUNTAIN FOREST AND RANGE EXPERIMENT STATION 

Forest Service 

U.S. Department of Agriculture 

Ogden, Utah 84401 

Robert W. Harris, Director

CONTENTS 

 Page 

INTRODUCTION ............................................................................................1 

INFORMATION PRODUCED........................................................................3 

INPUT VARIABLES .......................................................................................6 

COMPONENT MODELS ................................................................................8 

 Models for Development of Individual Trees.......................................8 

 Self-Calibration of Diameter Growth Functions...................................10 

STOCHASTIC FEATURES.............................................................................11 

 Random Error in Tree Development.....................................................12 

 Mortality ...............................................................................................13 

IMPLEMENTATION OF THE MODEL FOR LODGEPOLE PINE .............14 

 Diameter Change Model.......................................................................15 

 Equation for Predicting Height Increment............................................15 

 Model for Crown Ration Development ................................................16 

 Model for Mortality ..............................................................................16 

PROGRAM ORGANIZATION AND LOGIC SEQUENCES ........................20 

REFERENCES CITED.....................................................................................22 

APPENDIX I ....................................................................................................24 

 Field Plot and Computation Procedures for Describing 

 Competitive Status of Growth-Sample Trees .................................24 

 Relation of Size of Growth-Study Plots to Inventory 

 Plot Size ..........................................................................................27 

APPENDIX II ...................................................................................................28 

 Data Input for Program TREM∅D1.....................................................28 

APPENDIX III..................................................................................................31 

 Examples of Specification of Sample Design.......................................31 
ABSTRACT 

This paper describes a set of computer programs for combining quantitative silvicultural 

knowledge with past growth data from a sampled stand to make a prognosis of the course of 

development that the forest stand is expected to follow under alternative management 

prescriptions. An important design criterion of this procedure is that the prognosis model should 

apply to stands containing any mixture of species or age and size classes that grow as a 

community. The model simulates the deviation-amplifying aspect of the growth process by a 

unique procedure for introducing the stochastic elements in a deterministic computing algorithm. 

The growth rates predicted by the built-in models for diameter change are compared to the actual 

past growth of the sample trees to calibrate these models for the particular stand for which the 

prognosis is to be computed. Selection of trees to be cut at any period can utilize a variety of tree 

characters to emulate a wide range of silvicultural prescriptions. 

 An application of these programs to develop prognoses for lodgepole pine stands in the 

presence of an infestation of mountain pine beetles is described. 

 
INTRODUCTION 

 What are the future impacts of present management practices on tree growth? The 

answers to this question, for a variety of timber-growing sites and stand structures, encompass the 

full range of skills of the silviculturist. Answering the question in quantitative terms also requires 

the means to accomplish the large volume of computations that are necessary to represent the 

complex biological interactions that characterize the development of a forestry community. This 

paper describes a set of computer programs for combining our current silvicultural knowledge 

with past growth data form a sample stand to make a prognosis of the course of development that 

the forest stand is expected to follow under alternative management prescriptions. This model is 

deliberately termed a prognosis rather than either a simulation or projection. Most forest stand 

simulators begin with hypothetical distributions of trees in space; whereas this program begins 

with a diagnostic description of the present forest. The result is not a projection because the 

course of stand development depends on the detailed interaction of growth factors rather than 

upon following a initial trajectory implied by current growth rates. 

 For many of our forest types, our ecological and silvicultural knowledge is incomplete. It 

follows that the programs that comprise this model are also incomplete. My hope is that the 

design is sufficiently flexible that new capabilities can be added as silviculturists develop better 

quantitative representations of the various biological aspects of tree growth. Indeed the gaps in 

this model call attention to corresponding gaps in silvicultural research. 

 An important design criterion of this model is that the prognosis should apply to stands 

containing any mixture of species or age and size classes that grow as a community. That is, the 

model should apply equally well to pure even-aged stands or stands composed of a mixture of 

ages, species, and sizes. 

 Growth models that are tree-by-tree analogs of stand development can be useful adjuncts 

to silvicultural research. In this context, they can be used to interpolate among the limited 

number of treatment combinations that can be installed in a research study of feasible scope. For 

this use, resolution of the model should be capable of describing tree growth in sufficient detail to 

demonstrate subtle differences between treatments. On the other hand, in the context of forest 

management planning, much less detail is required about individual trees. Consequently, the 

development of this program has followed a middle course between the growth simulation 

models (described 
by: Mitchell 1968; Newnham and Smith 1964; lee 1967; and Arney 1972) that maintain the map 

coordinates of the trees through time and the stand projection programs such as TRAS (Larson 

and Gogorth 1970) that combine trees from widely diverse stands into common cells. The effect 

of competition from neighboring trees on the growth of an individual tree is retained through the 

early stages of projected time by maintaining the identity of the neighboring trees sampled at the 

same small diagnostic plot within the stand. In addition, variables expressing competitive status 

are computed from the relative position in the stand table and from the crown description 

(Appendix I) 

 Diagnostic examinations of forest stands are routine in the practice of silviculture. This 

program is designed to use sample data from these routine stand examinations as starting values 

for the prognosis. Two types of stand examinations are accommodated. In the first type, which is 

appropriate for surveying the regeneration phase of tree development, the sampling unit is an area 

of land such as a 4-milacre quadrat or a 1/300-acre circular plot. For such samples, the 

characteristics being modeled would be the species and heights of the dominant trees, a measure 

of competing vegetation, and little else. In the second type, the individual sample tree is the 

record unit. The tree characteristics recorded in this type of inventory emphasize the information 

needed to estimate its future course of development. As stands described initially by the stocked 

quadrat survey are projected through time by the program, the records are converted to the 

individual tree type of data for prognosis of subsequent stand development. 

 The functions that drive the prognosis are expressions for finite differences--that is, for 

the periodic rates of change of the various aspects of tree growth. Coefficients in the tree growth 

functions are estimated from past records of growth. Sources may include growth recorded in 

management inventories, and in research studies of silvicultural treatments for of insect and 

disease impacts. However, at the start of the prognosis, these coefficients based on prior analysis 

of growth are modified if the growth records of the stand being modeled provide sufficient 

evidence that the growth rated specified by the growth functions are not appropriate. Through 

this process of “self-calibration,” the model can accommodate itself to local peculiarities of site 

quality, genetic character, and tree vigor. In fact, the calibration variable can be interpreted as a 

measure of local site quality in healthy stands, or as a measure of impact of insect or disease 

outbreaks on the rate of accretion. 

 Growth is a process that amplifies the effects of previous departures from the mean 

growth level. To incorporate this characteristic of the growth process, special techniques of 

computation have been developed to retain the effects of the stochastic aspect of the growth 

process in the prognosis program. 

 The prognosis is developed by first estimating the changes to be expected in the tree 

conformation--diameter, height, and crown--during the next growth period. Then, the trees-peracre corresponding to each sample-tree record is reduced for the expected mortality rate 

appropriate to a tree of its characteristics growing in such an environment. The tree projection 

process is repeated for successive growth periods and appropriate displays of the stand’s 

development are produced for each period along the way. 

 Harvests (i.e. partial cuts, thinnings, or cleaning) can be scheduled at the start of any 

growth period. Selection of trees to be cut can utilize any of the characteristics describing the 

sample tree and the stand to simulate a variety of silvicultural prescriptions. If these silvicultural 

prescriptions are keyed to the timber classes and management alternatives specified for the 

Timber RAM Matrix algorithm (Navon 1971), then this growth prognosis model can be used to 

provide the yield schedules required by RAM for scheduling timber harvests. 
INFORMATION PRODUCED 

A surfeit of detail about the tree development is available within the computer data files. 

How to assemble this detail into information useful to the manager in compact form is no small 

problem. Certainly, numbers of trees, their distribution by size, the volume of scheduled 

harvests, and its species composition are essential elements of the output. Stand and stock tablesarrays of numbers of trees and volumes by diameter classes-are the customary means of 

presenting such information. However, diameter-class intervals appropriate to one stage of 

development of a stand would either provide excessive detail at later stages, or insufficient detail 

at earlier stages. Instead, the distribution of any attribute with regard to diameter is displayed by 

printing the lowest diameter such that a given fraction of that attribute for the stand is entirely 

contained in trees of that diameter and larger. For example, figure 1A is a portion of the output 

from a lodgepole pine stand as it is expected to develop from 1969 to the year 2020. In 1969 

there was a total of 509 trees per acre. Of these, 10 percent were larger then 8.9 inches d.b.h., 50 

percent were larger than 6.8 inches d.b.h., and 90 percent were larger than 5.7 inches d.b.h. At 

the same time, 50 percent of the total volume (which was 4,775 cu.ft./acre) was contained in 

stems larger than 7.3 inches d.b.h. 

 The periodic mean annual accretion (growth on surviving trees) and mortality from 1969 

to 1980 are indicated as 110 and 61 cubic feet per acre per year, respectively. The distributions of 

growth and mortality by size classes are provided with the same interpretation described for 

numbers of trees and volume. In 1990 the stand was thinned from below to a residual density of 

300 trees per acre. Volume removed was 535 cu.ft. 

 Species composition is indicated by displaying the percentages of the total volume (cu.ft.) 

that represent each of the three most plentiful species. Species composition is displayed for only 

those stand attributes that are measured in units of cubic feet. 

 Yields in units of merchantable product are, of course, vital to the utility of this program 

for management planning. Grosenbaugh (1954) and Bruce (1970) have shown how summaries of 

the primary units of volume (cu.ft.), bole surface area (sq.ft.) and bole length (ft.) can be used to 

predict yields when the wood is manufactured into a wide variety of products. Furthermore, the 

conversion from the trio of primary units to product yield is linear and additive. Consequently, 

sums of the volumes, surfaces, lengths, and numbers of stems can be accumulated by species over 

many stands in a population before the conversion in computed. 

 Growth trends of the individual trees of interest to supplement the stand summaries 

described above. By following individual tree records through the cycles of predicted growth, we 

can gain additional insight into the prognosis model. The second output table shows the 

development of the five sample trees that occur at the boundaries of the fractions of the initial 

diameter distribution (percentile). Species, d.b.h., height, crown ration, past diameter growth, and 

trees-per-acre associated with each of the five sample tree records are shown, along with the 

relative density of the stand (fig. 1B). 

 
1A STAND DATA CARD: 

2506316 4121 11210 2124329119006 62020 0 80 45 100 0.0 

STAND ID= 1 FAB=21 AREA =119 POINTS= 6 ELEV=43 S.I.= 80 AGE= 45

 FRACTION CF STAND 

YEAR ATTRIBUTE .9 .7 .5 .3 .1 TOTAL 

(LOWEST DBH IN STAND FRACTION) SF ECIES COMPOSITION

 1969 STAND- TREES 5.7 6.2 6.8 7.7 8.9 509. NO/A

 VOLUME 5.8 6.8 7.3 8.5 9.2 4775. CUFT 100.% LP, 0. % --, 0. % -

 SURFACE 39577. SOFT 

 LENGTH 31635. FEET

 ACCRETION 6.2 7.3 8.2 9.2 10.2 100. CUFT/YR. 100.% LP, 0. % --, 0. % -

MORTALITY 5.7 6.2 6.8 7.6 8.7 61. CUFT/YR. 100. % LP, 0. % --, 0. % -

 

 1980 STAND- TREES 6.1 6.8 7.4 8.4 9.6 426. NO/A

VOLUME 6.2 7.2 8.2 9.2 10.1 5203. CUFT 100. % LP, 0. % --, 0. %-

 SURFACE 40217. SOFT 

 LENGTH 29738. FEET

ACCRETION 6.7 7.7 8.8 9.8 10.9 98. CUFT/YR. 100. % LP, 0. % --, 0. % -

 MORTALITY 6.1 6.8 7.3 8.2 9.4 65. CUFT/YR. 100. % LP, 0. % --, 0. % -

 

 1990 STAND- TREES 6.4 7.2 7.8 9.1 10.3 364. NO/ A

 VOLUME 6.7 7. 7 8.8 9.8 I0.8 5536. CUFT 100. % LP, 0. % --, 0. % -

 SURFACE 40318. SOFT 

 LENGTH 27865. FEET

 REMOVAL 64. NO/A

 VOLUME 6.2 6.3 6.4 6.5 6.6 535. CUFT 100.% LP, 0. % --, 0. % -

 SURFACE 5123. SOFT 

 LENGTH 4526. FEET

 

RESIDUAL 7.0 7.6 8.4 9.4 10.5 300. NO/ A

 

 ACCRETION 7.7 8.6 9.6 10.5 11.7 88. CUFT/YR. 100. % LP, 0. % --, 0. %-

MORTALITY 7.0 7.6 8.1 9.1 10.1 62. CUFT/YR. 100. % LP, 0. % --, 0. % -

 

2000 STAND- TREES 7.5 8.1 9.0 10.1 11.1 257. NO/ A

 VOLUME 7.7 8.6 9.6 10.4 11.6 5258. CUFT 100.% LP, 0. % --, 0. % -

 SURFACE 34945. SOFT 

 LENGTH 21738. FEET

 

 ACCRETION 8.2 9.2 10.4 11.1 12.5 89. CUFT/YR. 100. % LP, 0. % --, 0. % -

 MORTAL ITY 7.5 8.1 8.8 9.7 10.7 66. CUFT/YR. 100. % LP, 0. % --. 0. % -

 

2010 STAND- TREES 7.9 8.7 9.6 10.6 11.9 221. NO/ A

 VOLUME 8.1 9.2 10.4 11.1 12.5 5493. CURT 100. % LP, 0. % --, 0. % -

 SURFACE 34440. SOFT 

 LENGTH 20098. FEET

 ACCRETION 8.7 9.9 11.1 11.8 13.4 87. CUFT/YR. 100. % LP, 0. % --, 0. % - 

 MORTALITY 7.9 8.5 9.3 10.5 11.5 70. CUFT/YR. 100. % LP, 0. % --, 0. % -

 

2020 STAND- TREES 8.3 9.2 10.2 11.3 12.8 189. NO/A

 VOLUME 8.6 9.8 11.0 11.7 13.3 5671. CUFT 100. % LP, 0. % --, 0. %-

 SURFACE 33577. SOFT 

 LENGTH 18420. FEET
1B STAND ID= 1 SAMPLE TREE RECORDS SITE= 80.0 

YEAR %TILE SPECI /

 NCHES) (FEET) RATIO GROWTH ACRE AGE DBH ACRE AREA REL DEN

ES DBH HEIGHT CROWN PAST DBH TREES/ STAN MEAN TREES BASAL STAND 

( I

1969 (10 YRS) . 9 L P 5 .7 53.0 33 0 .30 18.811 . 7 L P 6 .2 63.0 34 0 .30 15.899 . 5 L P 6 .8 67.0 35 0 .30 13.217 . 3 LP 7 .7 61.0 36 0 .40 10.308 . 1 LP 8 .9 65.0 38 0 .40 7 .716 45 7 .3 509.486 146.662 169.1 

1980 (11 YRS) . 9 L P 6 .0 60.4 33 0 .31 14.240 . 7 LP 6 .5 70.2 33 0 .35 12.506 . 5 L P 7 .2 74.2 34 0 .38 10.852 . 3 L P 8 .1 68.4 35 0 .44 8 .920 . 1 L P 9 .4 72.5 37 0 .53 7 .133 56 7 .8 426.398 142.766 158.9 

1990 (10 YRS) . 9 L P 6 .3 66.9 34 0 .29 11.145 

 .7 L P 6 .9 76.6 34 0 .31 10.088 . 5 L P 7 .5 80.6 35 0 .34 9 .041 .3 L P 8 .5 75.0 36 0 .39 7 .791 . 1 LP 9 .9 79.1 38 0 .47 6 .594 66 8 .7 300.005 124.406 132.4 

2000 (10 YRS) . 9 LP 7 .0 73.9 37 0 .68 0 .174 . 7 L P 7 .1 82.9 36 0 .28 7 .937 .5 LP 8 .2 87.3 37 0 .65 7 .348 . 3 LP 8 .9 81.5 38 0 .39 6 .680 . 1 LP 10.3 85.6 40 0 .41 6 .024 76 9 .3 257.446 120.780 125.3 

2010 (10 YRS) . 9 LP 7 .2 80.2 38 0 .26 0 .136 . 7 LP 7 .8 89.5 37 0 .63 6 .228 . 5 LP 8 .4 93.3 38 0 .22 6 .039 . 3 LP 9 .3 87.8 39 0 .35 5 .665 . 1 LP 10.8 92.1 40 0 .53 5 .467 86 9 .9 220.776 117.363 118.9 

2020 (10 YRS) . 9 LP 7 .6 86.5 43 0 .37 0 .105 . 7 L P 8:3 95.9 41 0 .49 4 .888 . 5 L P 8. 8 99.5 42 0 .39 4 .859 . 3 LP 9.9 94.4 43 0 .60 4 .757 . 1 L P 11.6 9P.7 44 0 .72 4 .920 96 10.5 188.809 113.379 112.3 

Figure 1.--Output for a pure LODGEPOLE pine stand to be thinned to 300 trees per acre in 1990. 
INPUT VARIABLES 

 The variables used to describe the stand at the start of the prognosis are listed below. 

Some of these items may be omitted if the growth functions for a particular forest type do not use 

them. A specific format for the input of these variables is given in Appendix II. 

Sample design (see Appendix III): 

 Stand expansion factor or sampling weight 

 Number of plots in the stand 

 Type of stand examination (quadrat description vs. tree enumeration) 

 Fixed-area-plot size 

 Variable-radius-plot basal area factor 

 Tree d.b.h. dividing fixed form variable plot tally 

 Management class code designating recently cut tree or recent mortality tree 

 Period for measurement of radial or height increment 
Site characters:

 Site index Aspect 

 

 Elevation Slope 

 Latitude Physiographic site 

 Habitat type Stockability1

 Competition from nontree species 

Stand characters: 

 Timber class Stand origin 

 Age (if even-aged) Total stand area 

 Proposed management prescription 

Tree characters: 

 Plot identification 

 Species 

 Number of trees represented by this record on the plot 

 D.b.h. 

 Height2 

 Live crown percentage3 

 Radial increment and bark thickness at b.h.3 

 Management class 

1

A.K. Wilson. Yield and productivity problems in Rocky Mountain States inventories. Intermt. 

For. and Range Exp. Stn., Ogden, Utah (In preparation) 

2

Can be included as a subsample. 

3

Can be omitted or included as a subsample. 
COMPONENT MODELS 

 Full implementation of this prognosis algorithm requires three kinds of models for: 

 Development of individual trees 

 Development of regeneration stands including ingrowth into existing stands 

 Transition from regeneration phase to individual tree phase 

Only the model for the first of these three processes is described in this paper. The latter two 

processes will require some additional silvicultural research before general models can be 

derived. Growth functions should be based on data derived from the area to which the model is 

to be applied. The self-calibration feature of this model only partially mitigates this admonition. 

Models for Development of Individual Trees4 

 The growth model for saplings and larger trees is a set of functions that predict the rate of 

increase of tree d.b.h., the rate of increase in tree height, the change in crown dimensions, and the 

change in bark thickness. The change in number of trees per acre represented by each sample tree 

is based on a function estimating the mortality probability. 

 The nature of the variables that should be included in the growth functions is controlled 

by the purposes for which the prognoses are to be used. For example, if the only course of 

development to be modeled is the unmanaged trends of natural stands, without catastrophic 

disturbance, then a variable representing past growth rate would be a very effective predictor of 

succeeding growth rates. It would include most of the effects of site, stand density, individual 

tree vigor, 

 4The logic of the program provides for distinct functions for 11 different species or 

species groups. To increase the number of species would not be particularly difficult. 
and more. But if we wished to model the effects of thinning or partial cutting, then the prognoses 

would be inadequate because the changes in stand density would not be reflected in changes in 

growth rates. The past growth variable would be an alias for stand density effects. Similar 

difficulties could arise from the combination of age and size in the prediction because size 

divided by age is, in effect, a measure of past growth. 

 The key growth function predicts the rate of increase of tree d.b.h. The dependent 

variable is the logarithm of the annual increase in the square of d.b.h. in inches; thus, this variable 

is equivalent to the logarithm of the basal area increase. Basal area was selected because its 

increase is most frequently linear with time. This linearity facilitates projection for intervals 

different from the growth interval over which the parameters of the model were estimated. If 

basal area increase is measured without bark, then the ration of basal area outside bark to basal 

area inside bark is used to convert the increment to outside-bark measure. 

 The logarithmic transformation is used here for two reasons: First, diameter growth rate 

distributions are bounded by zero at the lower end (ignoring the effects of bark sloughing) and so 

tend to be quite skewed; hence, the arithmetic mean is an inefficient estimator. Second, the 

variability of diameter growth rates about their mean tends to increase as the mean increases. The 

logarithmic transformation in most cases has resulted in a uniform variance. The method of 

Oldham (1965) is used to estimate the arithmetic mean of the diameter growth from the 

logarithmic model. A further refinement developed by Bradu and Mundlak (1970) was 

considered unnecessary because the standard errors of estimate are usually small enough that the 

differences between the two methods are trivial. 

 Predictor variables that are available include the site and tree characters listed previously. 

Additional variables that measure stocking or relative stand density such as crown competition 

factor, basal area, or stand density index can be computed from the distribution of the tree 

diameters. Variables that measure the competitive relations between trees in the stand include the 

percentile in the basal area distribution or the ration of tree d.b.h. to mean stand diameter 

(Appendix I). In addition, predictions for rates computed earlier in the sequence of calculations 

can be used as predictors. These models are summarized as follows: 

Tree growth components Predictor variables Data source

Annual basal area D.b.h., relative stand density, Increment cores, 

increment (b.a.i.) site, elevation, habitat type, remeasured plots 

percentile in basal area 

distribution, crown ratio 

Height increment Radial increment, habitat Stem analyses 

type, d.b.h., height 

Crown dimensions Relative density, percentile Temporary plots 

in basal area distribution, 

d.b.h. 

Bark ratio Same as b.a.i. Temporary plots or 

 tree samples 

Mortality rates Same as b.a.i. and radial Remeasured plots, 

increment plus pest population “last n years mortality”, 

models where applicable “years since death” 

 (truncated) 

 
 Procedures for estimating the functional forms and their coefficients are readily available 

in many texts on multiple linear or nonlinear regression for models with continuous dependent 

variables such as height and diameter increment, and for crown and bark dimensions. 

 The mortality models require rather different statistical techniques. One possible 

mortality model and an estimation procedure developed especially for that model have been 

describes by Hamilton (in preparation). Examples of how these models might be formulated are 

illustrated in a later section where the implementation of this prognosis procedure is described for 

lodgepole pine. 

Self-Calibration of Diameter Growth Functions 

 The self-calibration feature is intended to scale the diameter growth functions that are 

contained in the program so that the predictions match the actual growth rates measured on the 

trees in the stand to be modeled. First, the stand stocking that existed at the start of the period 

during which growth was recorded is estimated. To do this, the average basal area growth 

percentage is calculated for the growth-sample subtracting the product of present basal area times 

the mean ratio of past basal area increment to basal area derived from the growth-sample trees. 

The sum of diameters if similarly reduced using the square root of the ratio. Then, the stocking at 

the start of the period is computed from the number of trees, and the reduced sums of diameters 

and of their squares. Trees removed or dying during the calibration period are included in the 

prior stocking with no growth adjustment. All other stand parameters are assumed to have 

remained constant. 

 Deviations between predicted and recorded growth rates (scaled in units of the logarithm 

of change in the square of diameter) are then sorted and the median deviation calculated. The 

value of the median is subtracted from the constant term of the logarithmic growth function to 

calibrate it. Thus, the effect is to multiply each prediction by a correction factor. 

 The median was selected as the location parameter for the adjustment rather than the 

mean because it is less likely to be influenced by occasional outliers due to measurement errors or 

abnormalities of growth (Barrodale 196; Forsythe 1972). 
STOCHASTIC FEATURES 

 Random variation about a statistical mean is a characteristic of all growth phenomena. In 

modeling procedures, the primary objective is to produce estimates of future yields that are the 

expectations of the overall stand growth process. The approach that is generally used is to assign 

a random error drawn from an appropriate distribution to each prediction. 

 The nature of the distribution of the random component depends intimately upon the 

resolution of the estimation function with which the random variable is associated. In addition, 

the self-calibrating feature of this prognosis program influences the distribution of the 

unexplained variation that is to be represented by the random variable. For example, in the 

function for diameter change, there are variables that change from tree to tree; other variables 

change from period to period for the same plot and a few variables that quantify unchanging 

characteristics of the stand such as site, elevation, or habitat. Consequently, the unexplained 

variation about the regression surface will have three components; among trees, among periods, 

and among stands. The self-calibration procedure serves to remove the “among stand” 

component of the residual variation, leaving the other two components to be represented by the 

distribution of the random variable. 

 To appreciate the effect of resolution of the estimation function of the distribution of the 

random variable, consider two functions for diameter changes that differ only by including or 

excluding a variable that evaluates a significant effect of crown development. Crown 

development changes slowly with time; therefore, as an explanatory variable it has a high serial 

correlation from period to period for a single tree. A stand-growth model could use either 

function (assuming the simulation is not intended to compare pruning alternatives). However, if 

the function without the crown development variable were used, the variance of the random 

variable would have to be larger, and the serial correlation between the random variables assigned 

to each tree in successive periods would need to be larger than if the crown variable were 

included. Either alternative could be used to generate a stand simulation with the same expected 

values as long as the stand progresses in a way that does not modify the natural correlations 

between stand density and crown development. The difference in results between the two 

alternative formulations would show up only in the variability of repeated runs of the prognosis. 

The lower the resolution of the components, the greater would be the variability among repeated 

runs. 

 
 Which predictions should be subject to a random component and which can be held to 

their mean estimate is a choice that depends on the nonlinearity of the effects of the variation in 

subsequent calculations. If assessing the variability of the outcome per se were one of the 

objectives of the modeling, then most of the prediction equations would require a random 

component. To obtain the expected value of the random process, the whole sequence of 

computations would be repeated with different random errors and the process averaged over the 

replications. One of the drawbacks to this approach is that the volume of computations is very 

large if it is applied to the solution of all prediction equations that make up the overall model. 

Another drawback is that little is known of the serial correlation that ought to characterize the 

successive values of the random variables. Are large positive deviations from expected diameter 

increment more likely to be associated with large positive deviations from expected height 

increment--or with expected changes in the crown dimensions? 

 The approach used in the present version of this program may be considered in Monte 

Carlo terms, a “swindle.” The purpose is to produce a prognosis that overall is the result of 

averaging many replications of the random process without actually having to carry out the 

replications. 

Random Error in Tree Development 

 The program assigns all random effects to the distribution of errors of prediction of the 

logarithm of basal area increment. Basal area increment was selected to carry the stochastic 

variation because the effects of differing diameter growth rates ramify in highly nonlinear way 

through most of the remaining components and variables such as percentile in the basal area 

distribution, relative stocking, the height increment model, and the crown development model. 

This distribution is assumed to be Normal, with a mean of zero. The variance of this Normal 

Distribution is computed as a weighted average of two estimates; the first such estimate is derived 

from the regression analysis that developed the prediction function and the second estimate is the 

standard deviation of the differences between the actually recorded growth (transformed to the 

logarithm of basal area increment) for the sample trees in the population and their corresponding 

regression estimates. The weights assigned to these two estimates are 100 for the prior 

component of error, and the number of growth-sample trees in the stand for the second 

component of error (Mehta 1972) 

 The effects of modifying the predicted change in tree d.b.h. by a random variable can 

then be carried into other predicted changes in tree characteristics by using the change in d.b.h. as 

an independent variable in each successive model. The random variable associated with each tree 

record is saved until the following cycle so that the appropriate serial correlation can be presered 

in the distribution of the random variables. 

 The random component of change in tree d.b.h. is treated in two ways, depending on how 

many tree-records make up the stand being projected. When there are many tree records, the 

effects of any one random deviation on the growth rate of one tree would be blended with many 

other trees. Consequently, the stand totals should be quite stable estimates. Accordingly, a 

random deviate from the specified distribution is added to the logarithm of basal area increment. 

Because of the logarithmic transformation, the effect on predicted diameter increment is 

multiplicative. 

 When the stand is represented by relatively few sample trees, however, a different 

strategy is used. In order to increase the number of replications of the random effects, each tree 

record is augmented by two additional records. These new records duplicate all characteristics of 

the tree except the predicted change in d.b.h. and the number of trees per acre represented by the 

source tree record. The trees-per-acre value of the source record is reduced to 60 percent of its 

current value. The two new records are given 15 and 25 percent of the source value; thus, the 

three records together still represent the same number of trees per acre. 
 

Each of these three records is associated with one of the three portions of the error 

distribution characterizing the deviations about prediction (fig.2). The first record representing 

the 60 percent of the population (approximately the center of the distribution) is given a 

prediction to which is added the average value of the deviations in that portion of the normal 

distribution. This “biased” point is indicated by A in figure 2. The second record representing 

the upper 25 percent of the error distribution is given a prediction corresponding to point B, and 

likewise, the record for the lower 15 percent is given a prediction corresponding to point C. By

this method, the weighted average prediction for the three records is still unbiased. 

Which of these two procedures is followed is controlled by whether the number of tree 

records is greater than 1,350. Hence, a stand described by 150 tree records at the start would go 

through two projection cycles using the record-tripling approach before switching to the use of 

single random deviate for each record. 

Mortality 

Random fluctuations in mortality are notorious problems in analyzing forest growth. 

Unfortunately, our records that could be used to assess the distribution of the random variable for 

mortality are barely adequate to estimate an average mortality rate. The variability of mortality 

rates through time, in response to fluctuations in climatic stress, extreme winds, and other 

destructive agents is currently unavailable. Later versions of stand growth prognosis programs 

will, we hope, be able to assess the effects of the stochastic nature of mortality. 
IMPLEMENTATION OF THE MODEL 

FOR LODGEPOLE PINE 

 

 Lodgepole pine was the first forest type for which this growth prognosis model was 

implemented. This species was selected because it characteristically grows in pure, even-aged 

stands so that the development of the necessary component models would be simpler than would 

be the case for mixed stands. Data on diameter-growth rates were available for a wide range of 

stand densities from studies installed as part of the Intermountain Station’s silvicultural research 

on lodgepole pine. In addition, many lodgepole pine stands had been sampled in the course of the 

normal timber management planning inventories conducted by the Northern Region. 

Interregional site curves that include adjustments for stand density had been recently derived for 

this species (Alexander, Tackle, and Dahms 1967). In addition, a stand density study of 

lodgepole pine with very detailed data on crown development was also in progress in the 

silvicultural research project of the Pacific Northwest Station, Bend, Oregon (Dahms 1967). 

Lodgepole pine is not a particularly good forest type to demonstrate the full utility of the 

approach used in this growth prognosis program. Stand growth models such as developed for this 

species by Myers (1967) at the Rocky Mountain Station would be adequate for most purposes. 

However, stand growth models do not seem to be adequate for forest types of highly variable 

species composition or highly variable age class composition. Nor do stand growth models offer 

as much flexibility for comparing alternative silvicultural prescriptions as is possible with models 

treating individual tree records. Also, though the procedure may be overly detailed for a simple 

type such as lodgepole pine, the purposes for which this model has been developed would best be 

met by a unified approach applicable to all species and types. 
Diameter Change Model 

The data for estimating the change in d.b.h. are derived from three sources: The levels-ofgrowing-stock studies established in 1957 by David Tackle, formerly of Intermountain Station; 

the permanent sample plots established for management planning inventory on the Helena, 

Beaverhead, and Bitterroot National Forests; and the levels of growing stock studies established 

by the Pacific Northwest Station in the vicinity of Bend, Oregon. The steps followed in 

developing this model are described elsewhere.5

 The model is given by the following expression: 

where: 

 

Equation for Predicting Height Increment 

The data necessary to predict height increment from stand density, crown ratio, diameter 

increment, etc., were derived from data collected by destructive sampling in the course of 

management sampling inventories. Trees to be felled were selected by first drawing a random

sample of the inventory locations that have been measured. At each location a subsample of 

the trees was drawn so that ultimately the trees were selected with probability proportional to 

their basal area. In addition to the stand characteristics described by the standard management

planning inventory location data, the radial increment is measured on two radii of a trunk 

section at breast height. 

5

Dennis M. Cole and Albert R. Stage. Intermt. For. and Range Exp. Stn., Moscow, Idaho 

(Ms. In preparation) 
Then by using whorl counts, the height growth of the last 10 years is determined on the felled 

tree. From these data the following model is developed: 

where: 

Model for Crown Ratio Development 

The model used to predict changes in crown dimensions is based on the rate at which the 

base of the live crown recedes which is expressed as a function of the height increment and the 

current level of stand density. For stands in which the crown competition factor is less than 125, 

the rate at which the crown recedes is specified to be equal to one-fifth of the increase in height 

for the tree. For stands having a crown competition factor greater than 125, the crown is 

specified to recede at a rate of 0.61 times the height increment rate. 

A different model is used for trees for which no crown measurement was obtained. 

Under this alternative the height to the base of the live crown is predicted as a function of tree 

height, tree diameter, the relative position in the diameter distribution at the start of the prognosis, 

crown competition factor; and habitat type. The function is: 

where the variables are as defined previously with: 

Models for Mortality 

Two alternative models for mortality rates were developed for lodgepole pine stands. 

One of these is based on the mortality study of Lee (1971). The other model is based on the 

dynamic relation between a population of mountain pine beetle (Dendroctonus ponderosae

Hopk.) and the developing stand of lodgepole pine. The model for beetle-induced mortality was 

developed in cooperation with, and using data from studies by Walter E. Cole and Gene D. 

Amman, on file at Intermountain Station, Ogden. 
Endemic Mortality 

 Mortality rates were derived by Lee from yield tables for Alberta and verified by him 

using data from remeasured sample plots. Judging by the nature of their source, his rates are 

presumed to apply to the development of stands in the absence of mountain pine beetle. Overall 

rates are highest in stands composed of small-diameter trees. The rates decline with increasing 

mean d.b.h., reach a minimum at 10.6 inches, and then begin increasing. The rates are 

independent of stand density. 

 In order to distribute the mortality rates over the range of diameters within the stand, 

Lee’s rates were multiplied by a factor that depends on the percentile of the tree within the basal 

area distribution. The factor used was: 

 [0.25 + 1.5 (1. - PCT/100.)] 

where PCT is the percentile computed by the subroutine PCTILE which is described in 

Appendix I. 

 The effect of this factor is to give the lowest mortality rate to the tree of maximum d.b.h. 

in the stand, and to give the smaller trees a rate that increases as the percentile declines. The 

maximum rate for the smallest tree in the stand would be 1.75 times the average rate for the stand. 

The effect on introducing the PCT variable is to distribute the mortality more heavily among the 

smaller trees in the stand. Lee observed that the mean diameter of mortality trees was 2 inches 

less than the stand overall mean diameter, and that the distribution of the mortality appeared to 

follow the normal Gaussian distribution. Accordingly, he calculated the number of trees expected 

to die by diameter classes from the normal distribution. However, in his procedure, there would 

be no explicit relation to the number of trees actually in the class in a particular sample of the 

stand. The procedure using PCT provides and explicit estimate of the mortality rate for each tree 

record, thus overcoming the difficulty in Lee’s procedure. 

Beetle-Induced Mortality 

Mountain pine beetle infestations are a major cause of the disintegration of lodgepole 

pine stands. The severity of losses depends on the ecological habitat type and elevation (Roe and 

Amman 1970). The probability with which a severe beetle outbreak occurs is as yet unknown. 

However, one of the antecedent conditions of an outbreak is that the stand must contain some 

trees larger than 12 inches d.b.h. Within a stand, the emerging beetles attack trees with higher 

probability if the tree is in the upper end of the diameter distribution. In turn, the attack density 

(entrance holes per square foot of bole area) is higher on the larger, thicker trees (Safranyik and 

Vithayasai 1971). 

 The size of the emerging population depends on two dominating factors; the density of 

attack, and the thickness of the phloem within which the larvae feed and pupate (Amman 1969). 

Phloem thickness is directly related to radial growth. The quantitative relation of phloem to 

radial growth was established by the work of D. M. Cole6

 in the research work unit studying the 

silviculture of lodgepole pine. 

 Relative losses of beetles during flight are highest when the population density is highest 

and when the population is in the declining stages of an outbreak. 

6

Dennis M. Cole. Phloem thickness relationship in lodgepole pine trees. Intermt. For. and 

Range Exp. Stn. (In preparation.) 

 Rates for the various stages and aspects of the beetle population were estimated by Cole 

and Amman (1969) from historical records of beetle outbreaks. It remained only to combine 

these data with the model for the development of the tree population to generate a synthesis of the 

interaction of the pest and its host. 

 The logic of this mortality model is diagramed in figure 3. The cycle for the beetle 

population is 1 year in length. Annual tree mortality is deducted for each annual cycle. However, 

the other features of the trees are changed only at the end of the longer tree prognosis cycle. 

 A report being prepared by W. E. Cole and A. R. Stage7

 will provide more detail about 

the mountain pine beetle population model. 

7

W. E. Cole and A. R. Stage. Intermt. For. and Range Exp. Stn., Ogden, Utah 

(Ms. In preparation.) 
PROGRAM ORGANIZATION 

AND LOGIC SEQUENCES 

The growth prognosis program is organized as an interrelated set of subroutines executed 

under the control of a very brief main program. The first few subroutines serve to read in the 

control information, to set up the sampling probabilities for the trees and plots (NOTRE, see 

Appendix III), to fill in missing data (CRATET), and to calibrate the various growth projection 

functions (DGCALP). Once this initial housekeeping has been accomplished, the program 

proceeds to cycle through the projection intervals. The first major subroutine called within the 

projection cycle is TREGRO. The purpose of this subroutine is to monitor the growth of 

individual tree records, to bring in the aspects of random variation in diameter growth rates and to 

deplete the stand through the expected mortality and harvest. To accomplish this task four lower 

level subroutines are utilized: DGF calculates the diameter increments for each tree record; 

MORTS decreases the number of trees per acre represented by each tree record for its expected 

mortality; HTGF calculates height increment and crown changes; and CUTS contains the logic 

for selecting trees to be harvested for the management regime proposed for the stand. The next 

major subroutine to be called is STAND, a subroutine that summarizes the total stand attributes 

implied by the new tree records. This summary includes the number of trees per acre and their 

distribution by tree d.b.h., the volume characteristics of the total stand as summarized by the 

primary units of volume in cubic feet, surface area of the boles in square feet, and the 

accumulated total tree height. In addition, the total cubic foot volume representing the accretion 

on the initial tree population and the total cubic foot volume represented by the mortality are also 

summarized and their distributions shown by percentile of the diameter distribution. Also at this 

point in the computation cycle, predictions of stand growth expressed by growth projection 

formulae for the stand as a whole would be applied. However, at the present time this aspect of 
the projection logic has not been implemented. When the stand computations have been 

completed, a subroutine MERGE is called to merge the stand projections with the tree projections 

and to provide any feedback of information that would indicate a need to modify the individual 

tree growth projection functions or, alternatively, the stand growth projection functions. The final 

subroutine in the projection cycle is DISPLY that serves to bring out the displays of the growth 

prognosis and to accumulate the information on growth that is to be used as subsequent input to 

harvest scheduling algorithms. This completed the projection cycle and the flow of control in the 

program returns to TREGRO to start a new growth projection cycle. When all projection cycles 

for this management alternative have been completed, a new management alternative is 

introduced for the same stand and a new projection is started. 

 If the course of stand development and thinning schedule would be the same for several 

periods under more than one management regime, then the program would save the redundant 

calculations by starting the subsequent prognoses from the point where the regimes depart from 

one another. In turn, when all of the management alternatives have been completed, a new stand 

is introduced and the whole process is repeated. 

 In addition to these principal subroutines that contain the logic of the growth prognosis 

itself, there are numerous special subroutines which merely handle repetitive calculations such as 

sorting, ordering the data according to various attributed, determining percentile in the 

distribution of those attributes, and related computational details. Copies of the FORTRAN IV 

computer program and additional documentation can be obtained from the author: 

 

Albert R. Stage 

Forestry Sciences Laboratory 

 1221 South Main 

Moscow, Idaho 83843 
REFERENCES CITED 

Alexander, Robert R., David Tackle, and Walter G. Dahms 

 1967. Site indexes for lodgepole pine, with corrections for stand density. U. S. For. Serv. Res. Pap. 

 RM-29, 18 p. 

Amman, Gene D. 

 1969. Mountain pine beetle emergence in relation to depth of lodgepole pine bark. USDA For. Serv. 

 Res. Note INT-96, 8 p. 

Arney, James D. 

 1972. Computer simulation of Douglas fir tree and stand growth. Ph.D. Thesis, Oregon State Univ. 

Barrodale, I. 

 1968. L1 approximation and the analysis of data. Appl. Statis. 17:51-57 

Bella, I. E. 

 1971. A new competition model for individual trees. For. Sci. 17(3): 364-372 

Bormann, F. H. 

 1957. Moisture transfer between plants through intertwined root systems. Plant Physiol. 32(1): 48-55 

Bradu, D., and Y. Mundlak 

 1970. Estimation in lognormal linear models. J. Am. Statist. Assoc. 65:198-211. 

Bruce, David 

 1970. Predicting product recovery from logs and trees. USDA For. Serv. Res. Pap. PNW-107, 15p. 

Cole, Walter E., and Gene D. Amman 

 1969. Mountain pine beetle infestations in relation to lodgepole pine diameters. USDA For. Serv. Res. 

 Res. Note INT-95, 7 p. 

Dahms, Walter G. 

 1967. Low stand density speeds lodgepole pine tree growth. U.S. For. Serv. Res. Note PNW-47, 11 p. 

Forsythe, Alan B. 

 1972. Robust estimation of straight line regression coefficients by minimizing pth power deviation. 

 Technometrics 14(1): 159-166. 

 

Grosenbaugh, L. R. 

 1954. New tree-measurement concepts; height accumulation, giant tree, taper and 

 shape. USDA For. Serv., South For. Exp. Stn. Occas. Pap. 134, 32 p. 

Grosenbaugh, L. R. 

 1957. Point-sampling compared with plot-sampling in southeast Texas. For. Sci. 3:2-14. 

Hamilton, David A. 

 Nonlinear regression with a dichotomous dependent variable. Intermt. For. and Range Exp. Stn. 

 Ogden, Utah. (In preparation.) 
Jaakkola, Sipi 

 1967. On the use of variable size plots for increment research. Paper presented at Int. Union For. Res. 

 Organ., XIV Congr., Sect. 25:371-378 

Larson, Robert W., and Marcus H. Goforth 

 1970. TRAS- A computer program for the projection of timber volume. USDA For. Serv. Agric. 

 Handb. 377, 24 p. 

Lee, Yam (Jim) 

 1967. Stand models for lodgepole pine and limits to their application. Ph.D. Thesis, Univ. B.C. 

Lee, Y. (Jim) 

 1971. Predicting mortality for even-aged stand of lodgepole pine. For. Chron. 47(1): 29-32. 

Lemmon, Paul E., and F. X. Schumacher 

 1962. Stocking density around ponderosa pine trees. For. Sci. 8(4): 397-402. 

Mehta, J. S. 

 1972. On utilizing information from a second sample in estimating the scale parameter for a family of 

 symmetric distributions. J. Am. Statist. Assoc. 67(388): 448-452. 

Mitchell, K. J. 

 1968. Simulation of the growth of even-aged stands of white spruce. Diss. Abstr. 

 28B(10): 3951-3952 (Abstr.) 

Myers, Clifford A. 

 1967. Yield tables for managed stands of lodgepole pine in Colorado and Wyoming. U. S. For. Serv. 

 Res. Pap. RM-26, 20p. 

Navon, Daniel I. 

 1971. Timber RAM...a long-range planning method for commercial timber lands under multiple-use 

 management. USDA For. Serv. Res. Pap. PSW-70, 22 p. 

Newnham, R. M., and J. H. G. Smith 

 1964. Development and testing of stand models for Douglas fir and lodgepole pine. For. Chron. 

 40(4): 494-502. 

Oldham, P. D. 

 1965. On estimating the arithmetic means of lognormally-distributed populations. Biometrics 

 21:235-239 

Opie, Jack E. 

 1968. Predictability of individual tree growth using various definitions of competing basal area. For. 

 Sci. 14(3): 314-323. 

Roe, Arthur L., and Gene D. Amman 

 1970. The mountain pine beetle in lodgepole pine forests. USDA For. Serv. Res. Pap. INT-71, 23p. 

Safranyik, L., and C. Vithayasai 

 1971. Some characteristics of the spatial arrangement of attacks by the mountain pine beetle, 

 Dendroctonus ponderosae (Coleoptera:Scolytidae), on lodgepole pine. Appendix: statistical 

 analysis of the “hole-pairs” experiment. Can. Entomol. 103:1607-1625. 
APPENDIX I 

Field Plot and Computation Procedures for Describing 

Competitive Status of Growth-Sample Trees 

 

 Many of the growth functions that have been useful for predicting the growth rate of an 

individual tree include variables that describe the competitive status of the subject tree relative to 

the surrounding trees that make up the forest stand. Some of these methods require that the mapwise relations between trees be recorded in order to compute the measure of competition (Bella 

1971). Others might be as simple as a variable defined by the ration of the subject tree d.b.h. to 

the root-mean-square d.b.h. of all trees in the stand. In my studies of growth of grand fir, and 

studies with D. M. Cole, of Intermountain Station, on the growth of lodgepole pine, it appears 

that if the crown development of the subject tree and its relative position within the diameter 

distribution of the stand are both included in the statistical model, then there is little added 

explanatory value to be derived from variables calculated from the map-wise distribution of 

stems. These studies admittedly were based on trees growing in undisturbed stands where the 

crowns had evolved in relation to local variations in stand density. However, even in stands 

where the equilibrium has been disturbed, there is evidence for tree-soil moisture relations that 

would permit trees to benefit from decreased competition for moisture in parts of the stand farther 

from the subject tree than is usually considered in competition models based on stem distributions 

is space (Bormann 1957). This view is supported by the analyses of subject tree growth showing 

that as the zone of competitive influence is increased, the explanatory power of the tree-centered 

measure of competition is increased (Opie 1968; Lemmon and Schumacher 1962). 

 The variable available in this prognosis program to represent the competitive status of the 

ith sample-tree record in relation to its surrounding stand are either: 

DBH (I) /RMSQD 

or 

PCT (I) 

where 

DBH (I) = Diameter of the ith tree in inches 

RMSQD = Diameter of the tree of mean basal area in inches 

PCT (I) = Percentile in the basal area distribution 
For example, PCT (I) = 100 implies that the ith tree is the tree of largest diameter in the stand. 

PCT (I) = 75 would correspond to the tree whose position in the diameter distribution was such 

that 25 percent of the stand basal area was in trees having a larger d.b.h. than the ith tree, and 75 

percent was in trees having a diameter less than or equal to the ith tree. Percentile in the basal 

area distribution has a very simple interpretation when sampling with a wedge prism or other 

angle gage. If 8 trees were found to qualify at a point sample, then a ranking by diameter would 

permit one to calculate the percentile simply as 100 for the largest tree, 100 * 7/8 = 87.5 for the 

second largest, 100 * 6/8 = 75 for the third largest, etc. If several points are tallied in the same 

stand, the procedure would be the same. For the ith ranked tree, in the sample of N trees, 

percentile could be calculated as: 

PCT (I)=(N-I +1)/N*100 

 When the calculations are based on a stand table, or a list of sample trees having varying 

numbers of trees-per-acre associated with each tree record, then two computer subroutines can 

facilitate the computation of percentile. The first routine, IDSORT, is a general sorting program 

that indexes an array of tree records according to a decreasing sequence of d.b.h.’s. The second 

routine, PCTILE, uses the indices form IDSORT to calculate the percentile in the distribution. 

These two routines are listed in figure 4. 

 The use of these two routines is illustrated in the following segment of the FORTRAN IV 

program: 

 C N = NUMBER OF TREE RECORDS 

 C DBH = ARRAY OF DIAMETERS OF SAMPLE TREES 

 C FNO = ARRAY OF NUMBER OF TREES PER ACRE CORRESPONDING TO THE DBH 

 C ARRAY 

 C IND = ARRAY OF INDICES INDICATING ORDER OF SIZE (DBH) 

 C BA = ARRAY OF BASAL AREAS/ ACRE 

 C PCT = ARRAY OF PERCENTILE CORRESPONDING TO DBH ARRAY 

 C TOT = TOTAL STAND BASAL AREA 

 INTEGER * 2 IND 

 DIMENTION DBH (N), FNO (N), PCT (N), BA (N), IND (N) 

 DO 1 J = 1, N 

 1 BA (J) = DBH (J) * DBH (J) * FNO (J) * .0054541 

 CALL IDSORT (N, BA, IND) 

 CALL PCTILE (N, IND, BA, PCT, TOT) 

PCT and TOT are assigned values by PCTILE that can be utilized along with other descriptors of 

the tree to produce data for building growth or mortality prediction models by statistical 

estimation procedures. 
 SUBROUTINE IDSORT (N, CHAR, IND) 

C SHELL SORTING ALGORITHM PROGRAMMED BY ROBERT M. RUSSELL 

 INTEGER * 2 IND (N) 

 DIMENSION CHAR (N) 

 IF (N. LE. 0) GO TO 100 

 DO 2 J = 1, N 

 2 IND (J) = J 

 IF (N. EQ. 1) GO TO 100 

 M = 1 

 20 M = M + M 

 IF (M. LT. N) GO TO 20 

 M = MAX0 (1, M/2 – 1) 

 30 I = 1 

 IM = I + M 

 40 J = I 

 JM = IM 

 I HOLD = IND(IM) 

 50 JR = IND(J) 

 IF (CHAR(JR). GT. CHAR (IHOLD)) GO TO 60 

 IND (JM) = IND (J) 

 JM = J 

 J = J – M 

 IF (J. GT. 0) GO TO 50 

 60 IND (JM) = IHOLD 

 I = I + 1 

 IM = I + M 

 IF (IM. LE. N) GO TO 40 

 M = M/2 

 IF (M. GT. 0) GO TO 30 

 100 RETURN 

 END 

 SUBROUTINE PCTILE (N, IND, CHAR, PCT, TOT) 

C COMPUTES PERCENTILE WITHIN THE DISTRIBUTION OF CHAR SUCH THAT 

C THE LARGEST INDIVIDUAL HAS PCT(I) = 100 

C IND = INDEX TO SORTED ORDER OF CHAR FROM SUBROUTINE IDSORT 

 INTEGER * 2 IND 

 DIMENSION IND(N), CHAR(N), PCT(N) 

 NMI = N – 1 

 PCT (IND(N)) = CHAR (IND(N)) 

 DO 10 I = 1, NM1 

 J = N-I 

 PCT (IND(J)) = PCT (IND(J+1)) + CHAR (IND(J)) 

 10 CONTINUE 

 TOT = PCT (IND(1)) 

 PCT (IND(1)) = TOT / 100 

 IF (TOT. LE. 0.0) RETURN 

 DO 20 I = 2, N 

 PCT (IND(I)) = PCT (IND(I)) / PCT (IND(1)) 

 20 CONTINUE 

 PCT (IND(1)) = 100. 

 RETURN 

 END 

Figure 4—FORTRAN IV coding of IDSORT and PCTILE 
Relation of Size of Growth-Study Plots to Inventory-Plot Size 

 When small plots (or point-samples with large basal-area factors) are used to sample the 

irregular spacing of trees that characterizes most forest stands, the estimates of stand density 

show a wider variation than would occur if larger plots were used. Grosenbaugh and Stover 

(1957) discuss how small-plot estimates are related to larger, concentric plots through a 

distribution that has a skewness that changes with density. Jaakkola (1957) suggests that the size 

of the sample plot used in stand density studies may affect the magnitude of regression 

coefficients associated with stand density. Intuitively, there must be some optimum plot size that 

depends of tree size for explaining the effect of stand density on tree growth. If so, then variable 

plots should be better for explaining this effect than fixed-area plots if a wide range of tree sizes 

are to be sampled with the same plot design. In addition, there must be some optimum basal-area 

factor for sampling the stand density variable in tree growth studies. Unfortunately, it is also 

likely that this optimum size will vary by site and species. 

 An important feature of this growth prognosis algorithm is that the effect of the bias that 

may be present in any of the diameter-growth models is compensated by the self-calibration 

features that scale the median and standard deviation of the residuals in relation to the past 

performance of the stand as it was actually sampled. If a regression coefficient is too small in 

absolute magnitude, then the residual variation will be larger, and the scaling of the stochastic 

multipliers will modify the estimates accordingly. 

 Local variation in stand density is introduced in the same manner. If a sample tree is 

growing in an area that deviates from the average density, then its contribution to the standard 

deviation of residuals will be large, and the effect will persist in variation of predicted diameter 

increments. However, the resolution of the overall model will be best if the field inventory plot 

design and the plot design of the growth studies are as similar as possible and both designs are 

capable of reflecting local conditions that affect tree growth. 
APPENDIX II 

Data Input for Program TREM∅D1 

 Codes for species, habitat type, etc., are defined in a BLOCK DATA subroutine that can 

be readily modified. Definitions and units for other variables such as the index age for site only 

need to agree with the corresponding variables used in the growth models. 

Data cards needed: 

 #1 & 2) The stand information format cards. These will contain the format (enclosed in 

parentheses) needed to read in the variables on card #7. 

 #3 & 4) The tree record format cards. These will contain the format (enclosed in 

parentheses) needed to read in the variables on card # 8. 

 #5) Contains the following variables: 

Variables Columns Type and format Definition 

NPLT 8-15 Alphanumeric 2A4 Stand Identification 

NCYC 16-18 Integer 13 Maximum number of cycles 

IY (J) 19-21, 22-24, Integer array The excess of the calendar year over 1900 

 . . . 2013 for successive projection cycles for 

 J = 1,..., NCYC+1. 1900+IY (1) = year of 

 initial inventory. 

 #6) Contains the following variables: 

Variables Columns Type and format Definition 

ALPH 8-15 Alphanumeric 2A4 Stand Identification 

CTN (J) 16-20, 21-25, Real array 11F5.0 Residual stand density to be left after 

 . . . thinning at the start of each projection 

 cycle. (If CTN (J) equals or exceeds the 

 stand density, no thinning will be per- 

 formed. 
 #7) Contains the following variables to be read in under the format provided on data 

 cards #1 and #2: 

Variable Type Definition 

S1 Alphanumeric Space holder8 

S2 Alphanumeric Space holder 

IF∅R Integer Forest 

IBLK Integer Block  

ICPT Integer Compartment  = NPLT on card #5 

ISBCPT Integer Subcompartment  ALPH on card #6 

ISTAND Integer Stand  

S3 Alphanumeric Space holder 

IS∅ Integer Stand origin 

S4 Alphanumeric Space holder 

IBAP Integer Basal area factor for variable radius plots 

S5 Alphanumeric Space holder 

IPT Integer Number of points sampled in stand 

ITYPE Integer Habitat type 

S6 Alphanumeric Space holder 

IAREA Integer Total stand area 

IPC Integer Photo interpretation class 

S7 Alphanumeric Space holder 

S8 Alphanumeric Space holder 

S9 Alphanumeric Space holder 

IELEV Integer Elevation in 100’s of feet above sea level 

S10 Alphanumeric Space holder 

S11 Alphanumeric Space holder 

S12 Alphanumeric Space holder 

ISL∅PE Integer Slope, scaled 0 to 9 

IASPCT Integer Aspect, scaled 0 to 8 

S13 Alphanumeric Space holder 

 8

Space holders have been inserted in the read-list to include all the fields designated on USDA 

Forest Service form R1-2410 (5/72). 
Variable Type Definition 

IPSITE Integer Physiological site class 

ISITE Integer Measure of site quality or index 

IAGE Integer Age of the stand in years 

IFINT Integer Number of years for which past tree 

Growth has been measured 

IDG Integer = 0 if past diameter growth is read into 

  DG on card #8 

 = 1 if a past diameter growth is read into DG 

SAMPR Real Sampling probability (inverse weight) for 

Stand (see Appendix III) 

 #8) Contains the following sample tree variables to be read in under the format provided 

 on data cards #3 and #4: 

Variable Type Definition 

IPLT (2) Alphanumeric Stand identification 

ITRE (I) Integer Point identification 

PROB (I) Real Number of trees on plot represented by this 

 tree record 

ITH Integer Tree history 

ISP (I) Alphanumeric Species code 

DBH (I) Real Diameter breast height in inches 

DG (I) Real Periodic diameter growth in inches or past 

 d.b.h. depending on IDG on card #7 

HT (I) Real Tree height in feet 

ICR (I) Integer Crown ratio 

IDCD Integer Damage code 

IMC (I) Integer Management code – tree or cover class 

 #9) Like card #8 

 •

 •

 #K-1) Like card #8 

 #K) SEND to signify end of stand, in same columns as the left four placed of IPLT 

 #K+1) Repeat sequence of cards #5 to K as many times as needed for succeeding stands 

 . ) 

 #n) Columns 8-11 contain SEND to signify end of input data. 
Appendix III 

Examples of Specification of Sample Design 

 To illustrate the use of the variables IBAP and IPT, several cases will be described that 

involve changes in some constants established in the BLOCK DATA subprogram. 

Case 1 

 A stand tallied at a set of 12 points. At each point, trees less than 5 inches d.b.h. are 

recorded on a 1/300 acre fixed plot. Trees greater than or equal to 5 inches d.b.h. are tallied on a 

variable plot established by an angle gauge having a basal-area factor of 20 square feet/tree. 

Then: 

 IPT = 12 on card #7 

 IBAP can be set to 20 or 0. The latter will default to BAF = 20 established in BLOCK 

 DATA. 

Case 2 

 Same as for Case 1, except the variable plot has a basal area factor of 40 square feet/tree. 

Then: 

 IPT = 12 

  on card #7 

 IBAP = 40  
Case 3 

 A stand tallied at a set of 16 points. At each point, trees less than 3 inches d.b.h. are 

recorded on a 1/256 acre plot. Trees greater than or equal to 3 inches d.b.h. are tallied on a 

variable plot established by an angle gauge having a basal-area factor of 40 square feet/tree. 

Then: 

 FPA = 256. 

 BRK = 3.0  established by recompiling BLOCK DATA 

 BAF = 40.0 

 IPT = 16 

  on card #7 

 IBAP = 0 

Case 4 

 All trees are tallied on 10 fixed-area plots of 1/5 acre. Then: 

 FPA = 5. (Reciprocal of 1/5 acres) 

  established by recompiling BLOCK DATA 

 BRK = 9999.  

 

 IPT = 10 on card #7 

Case 5

 Plot sizes and numbers vary from stand to stand, but all trees are tallied on the same sized 

plot at any one sample point. Then: 

 FPA = 1  

  established by recompiling BLOCK DATA 

 BRK = 9999. 

 IPT = 1. 

 n  on card #7 

 SAMPR = - ∑ ai  

 i = 1 

where ai equals the area n acres of each of the n sample plots in the stand. 

 Furthermore, if the sampling probability changes arbitrarily from tree to tree in the stand, 

PROB (I) on card type #8 can be set to the reciprocal of the sampling probability for that tree. In 

this way, it is possible to accommodate virtually any sampling design including 3-P samples. 

U. S. GOVERNMENT PRINTING OFFICE: 1973_784-293/ 6 REGION NO. 8 
ALBERT R. STAGE

1973. Prognosis model for stand development. USDA For. Ser.

Res. Pap. INT-137, 32 p., illus. (Intermountain Forest &

Range Experiment Station, Ogden, Utah 84401.)

Describes a set of computer programs for developing prognoses of the

development of existing stand under alternative regimes of management. Calibration techniques, modeling procedures, and a procedure

for including stochastic variation are described. Implementation of the

system for lodgepole pine, including assessment of losses attributed to

an infestation of mountain pine beetle, is described.

___________

OXFORD: 5:.1 KEYWORDS: increment forecasting, yield regulation, stand projection, tree growth, modeling, lodgepole pine, mountain pine beetle impact, simulation.

ALBERT R. STAGE

1973. Prognosis model for stand development. USDA For. Ser.

Res. Pap. INT-137, 32 p., illus. (Intermountain Forest &

Range Experiment Station, Ogden, Utah 84401.)

Describes a set of computer programs for developing prognoses of the

development of existing stand under alternative regimes of management. Calibration techniques, modeling procedures, and a procedure

for including stochastic variation are described. Implementation of the

system for lodgepole pine, including assessment of losses attributed to

an infestation of mountain pine beetle, is described.

___________

OXFORD: 5:.1 KEYWORDS: increment forecasting, yield regulation, stand projection, tree growth, modeling, lodgepole pine, mountain pine beetle impact, simulation.

ALBERT R. STAGE

1973. Prognosis model for stand development. USDA For. Ser.

Res. Pap. INT-137, 32 p., illus. (Intermountain Forest &

Range Experiment Station, Ogden, Utah 84401.)

Describes a set of computer programs for developing prognoses of the

development of existing stand under alternative regimes of management. Calibration techniques, modeling procedures, and a procedure

for including stochastic variation are described. Implementation of the

system for lodgepole pine, including assessment of losses attributed to

an infestation of mountain pine beetle, is described.

___________

OXFORD: 5:.1 KEYWORDS: increment forecasting, yield regulation, stand projection, tree growth, modeling, lodgepole pine, mountain pine beetle impact, simulation.

ALBERT R. STAGE

1973. Prognosis model for stand development. USDA For. Ser.

Res. Pap. INT-137, 32 p., illus. (Intermountain Forest &

Range Experiment Station, Ogden, Utah 84401.)

Describes a set of computer programs for developing prognoses of the

development of existing stand under alternative regimes of management. Calibration techniques, modeling procedures, and a procedure

for including stochastic variation are described. Implementation of the

system for lodgepole pine, including assessment of losses attributed to

an infestation of mountain pine beetle, is described.

___________

OXFORD: 5:.1 KEYWORDS: increment forecasting, yield regulation, stand projection, tree growth, modeling, lodgepole pine, mountain pine beetle impact, simulation.
Headquarters for the Intermountain Forest and 

Range Experiment Station are in Ogden, Utah. 

Field Research Work Units are maintained in: 

 Boise, Idaho 

Bozeman, Montana (in cooperation with 

 Montana State University) 

Logan, Utah (in cooperation with Utah 

 State University) 

Missoula, Montana (in cooperation with 

 University of Montana) 

Moscow, Idaho (in cooperation with the 

 University of Idaho) 

Provo, Utah (in cooperation with Brigham 

 Young University) 

Reno, Nevada (in cooperation with the 

 University of Nevada)