r 

Prediction of Height Increment 

for Models of Forest Growth 

Albert R. Stage 

This file was created by scanning the printed publication.

Errors identified by the software have been corrected;

however, some errors may remain.

USDA Forest Service 

Research Paper INT-164 

January 1975 

Prediction ofHeight Increment 

for Models of Forest Growth 

Albert R. Stage 

INTERMOUNTAIN FOREST AND RANGE ·EXPERIMENT STATION 

Forest Service 

U.s. Department of Agriculture 

Ogden, Utah 84401 

• Roger R. Bay, Director 
THE AUTHOR 

ALBERT R. STAGE is principal mensurationist at the Forestry Sciences 

Laboratory, Moscow, Idaho. His research has included studies of 

planning methods for forest management, forest inventory techniques, site evaluation, and methods for estimation of stand growth 

and yield. He is Project Leader of a research team studying the 

problems of intensive culture of the cedar-hemlock-grand fir ecosystem and problems of timber management planning. 

ii 
,•;; 

CONTENTS 

INTRODUCTION. o o • o o 

DESCRIPTION OF DATA. 

DERIVING THE FUNCTIONAL FORM FOR 

THE PREDICTION EQUATIONS • 

Differential Model • o • • 

Finite Difference Model 

Differential Model--Logarithmic Form • 

Removing Bias in Logarithmic Form o 

APPLICATIONS •. o o o •• o • • o o •• 

Calculating Current Annual Increment .••• 

LITERATURE CITED. • . • . . • . ••. 

APPENDIX: FORTRAN IV Subroutine • 

iii 

Page 

1 

2 

8 

9 

10 

11 

15 

16 

16 

18 

19 
ABSTRACT 

Functional forms of equations were derived for predicting 10-year periodic 

height increment of forest trees from height, diameter, diameter increment, 

and habitat type. Crown ratio was considered as an additional variable for prediction, but its contribution was negligible. Coefficients of the function were 

estimated for 10 species of trees growing in 10 habitat types of northern Idaho 

and northwestern Montana. A procedure using these coefficients for calculating 

the components of current annual volume increment from diameter, height, and 

past radial increment is described, and its FORTRAN implementation listed. 

iv 
INTRODUCTION 

How fast do trees grow in height? The answer requires a great deal of detail about 

the tree's environment and about the tree itself, or some specific growth measure on the 

tree that indicates the combined effects of all environmental factors and the tree's 

individual characteristics. Diameter increment is such a measure. It is much more 

easily measured than height increment, and responds to the same growth determinants. 

Diameter increment, however, is much more responsive to effects of stand stocking 

than is height increment. Past effects of stocking on height growth relative to diameter growth are indicated by the tree's form, as measured by the height/d.b.h. ratio, 

and by its crown ratio (live crown length/total height). In this report, we seek to 

develop a prediction equation that relates height increment to concurrent diameter 

increment, tree height,, diameter, crown ratio, and habitat type (Daubenmire and 

Daubenmire 1968) . 

To use this equation for calculating the height increment component of current 

volume increment of trees on inventory plots, a computer subroutine is provided in 

the appendix. For more extended prognoses, the same equation for predicting height 

increment can be imbedded in a computer model of forest stand development (Stage 1973) . 

For many years Forest Survey in the Intermountain Forest and Range Experiment 

Station has calculated current volume increment from changes in height computed from 

diameter increment and diameter. Wright (1961) recommended a similar procedure. Both 

approaches were derived from a single curve of total height over d.b.h. by determining 

the differences in the curve height for current d.b.h. and for d.b.h. plus diameter 

increment. Hence, these approaches assume that all trees of a particular species follow height/diameter curves that have the same slope for a given d.b.h. 

Better prediction equations should be possible where data are available from 

direct measurements of height increment. Prediction equations presented in this report 

are based on analysis of 1,165 trees felled as part of management planning inventories 

of the Kaniksu, Coeur d'Alene, St. Joe, and Lewis and Clark National Forests in northern 

Idaho and western Montana. 

1 
DESCRIPTION OF DATA 

Ten species distributed among 10 habitat types are represented in the data 

(table 1). Distribution of samples with respect to height and d.b.h. are shown by 

species in figure 1, and by habitat types in figure 2. Periodic diameter growth 

(inside bark), ~D, was measured concurrently with periodic height growth, ~H, for the/ 

most recent 10-year period. Values for height (H) and diameter at breast height (D)~ were determined as they existed at the start of the increment period because the regression estimate is required to predict the change in height as a function of height, 

diameter, and projected diameter increment (inside bark). Of necessity, the crown ratio 

used was observed at the end of the growth period. Fortunately, crown ratio changes 

very slowly within 10 years. 

The trees were measured during the inventories of 1969-1972. Consequently, their 

most recent 10-year growth is influenced by the climate and other temporal events from 

about 1959 to 1972. 

lfsark increment was not deducted when past d.b.h. was calculated. The slight 

overestimate of past d.b.h. can cause only a slight bias in the estiamte of height 

growth. 

2 
Table I.--Distribution of sample trees by species a:nd habitat type 

Habitat 

"-

(j 

~ "- N 

N ~ (j ~ Ci;) 

"- ~ C) Ci;) ~ ~ ~ ~ ~ 

"-~ "-.C!;) ·~ Ci;) ~ (j (j (j (j 

"-~ ·~ C) ~ g ~ ~ ~ (j (j (j ;:s (j Ci;) '\j(j (j N (j C)(j C) C) 

t!J~ 

~~ ~C) ~ E: E: ~E: ·~ E: ·~N ·~ (j ;:s·~ ·~ § ;:s N (j·~ ·~ ~·~ Ci;)·~ C!;)N C!;)·~ 

C!;) N Ci;) (j -i:S~ N~ ~ ~~ (j~ (j~ (j Ci;) (j·~ 

~C) ~ ~ \:)JC!;) 

"-.·~ ~Ci;) NC!;) N~ N~ N~ 

C)~ C) C) 

~~ ·~ ·~ · ... ~ ~ ·~ ·~ 

]~ '\3C!;) Ci;)~ (j~ (j~ Ci;)~ C!;)C) Ci;) ~ Ci;) ~ 

;:s ~ ;:sN ~ ~ ·\"""";)~ ~~ ~ ~ ~N ~ ~ ~ ~ 

~~ ~~ ~ (j ·~ (j ;:s(j ;:s (j ·~ (j ·~ ~ ·~ ~ ·~ (j 

S;eecies ~t() Ci;)~ C!;)\..) ~~ ~~ ~~ ~~ ~N ~~ ~~ 

~ ~ ~ ~ ~ ~ ~ Total 

Ponderosa 

pine 12 16 8 36 

Douglas-fir 108 53 28 132 13 40 16 6 6 402 

Western larch 7 17 9 27 4 14 2 80 

Lodgepole 

pine 6 49 14 3 17 1 8 88 186 

Grand fir 49 3 76 1 129 

Western white 

pine 22 2 76 1 1 102 

Western 

redcedar 1 7 39 47 

Western 

hemlock 1 104 8 113 

Engelmann 

spruce 2 3 6 3 14 

Subalpine fir 5 13 32 5 1 56 

Total 108 78 79 251 46 387 43 67 8 98 1,165 

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Figure 1.--(con.) 

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Figure 2. Distribution 

of sample trees by 

habitat type~ height~ 

and diameter. 

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DERIVING THE FUNCTIONAL FORM 

FOR THE PREDICTION EQUATIONS 

A set of height/diameter curves for three even-aged stands of grand fir is shown 

in figure 3. These curves are plotted solutions of a nonlinear equation for predicting 

tree height from diameter, site quality, and age of the stand. Analysis of these data 

confirmed that the allometric relation: 

ln(H) = a + b ln(D) ( 1) 

does indeed apply to the development of trees as age increases, but that the coefficients 

a and b depend upon the competitive status of the trees. Competitive status can be 

defined by the distance of the (H~D) point for a particular tree from th~ curve of 

relation (1) for dominant trees (Perkal and Battek 1955). The dashed' line shows the 

trend followed by the average for the dominant and codominant trees in the stand. 

Trees in lower crown classes move more steeply upward. Conversely, dominants move 

along lines of lower slope. 

110 

100 

80 

/---In H =2.447 +0.81n D 

46 Ql 

Ql 

I- 60 I 

~ 

w 

I 

40 

19 

20 

DIAMETER (b.h. inches) 

Figure 3. Sequence of height/diameter curves for even-aged stands of grand fir. 

8 
Instantaneous growth rates can be obtained by taking the differential of 

equation (1): 

'dH 

or, in its logarithmic form: 

b !i aD D 

ln(aH) = ln(aD) + ln(H) - ln(D) + ln(b) 

(2) 

(3) 

On the other hand, periodic growth rates expressed as a finite difference derived 

from (1) would be: 

which is the same as: 

(4) 

where the subscripts 1 and 2 designate measured values at the start and end of growth 

period, respectively. 

Models were derived from (2), (3), and (4) by using various transformations of 

height, the ratio of height to diameter, and crown ratio to estimate the b parameter 

for each species. The best transformations of these independent variables were selected 

by combining them in groups of sets for screening overall combinations ~f the groups 

by using one set from each group (Grosenbaugh 1967). Coefficients in these alternative 

models were estimated by least-squares regression. Goodness-of-fit indices (Furnival 

1961) for the several transformations of the dependent variable were compared using the 

best regression for each transformation of the dependent variable. 

For the screening of these alternative models, data from 909 trees of 10 species 

in the northern Idaho forests were used. 

Differential Model 

The dependent variable in this model is the 10-year periodic height increment, 6H. 

Some coefficients (cj) in the model may be different for each species; other coefficients may be constant for all 10 species. Those that vary with species have an additional subscript i. 

H 6D 6D 2 D 6H = D 6D [c 1i + c2i ln (CR) + c3 D + c4 (D) + cs ln (H) + c6i H6D] 

The expression in the brackets represents the estimate of bin equation (2). In computation, c6. was the set of constant terms in the regression model. 

1--

Table 2 is an analysis of variance showing the improvement in the regression sumof-squares for successively more complex collections of variables. In this table, for 

example, comparison level 2 indicates that when either the coefficients of H6D/D or 

the constant terms are allowed to vary by species, then the fit is improved by a significant amount. Of these two alternatives, the former has a slight advantage. At 

comparison level 3, varying constant terms by species is of little value so long as the 

coefficients of H6D/D depend on species. However, the variable ln(H) is shown to be 

needed to represent the effect of changing height on the b coefficient. At comparison 

l~vel 4, crown ratio is still of little use, but addition of 6D/D and its square 

results in a considerable improvement. 

9 
llt 

Table 2.--Analysis of improvement of regression model attributable to adding variables. 

Dependent variable = ~H 

Proportion of 

Comparison : remaining variance Marginal : Mean 

level Source d. f. explainedV : sum of squares~/ : square F 

1 

[ ~~D/D ] 1 49469.808 

1 2263.450 2263.450 263.0 

2 [ H~D/D. J 9 1..-Sp 0.065 776.592 86.288 10.0 

1. 1..-Sp 9 .060 728.725 80.969 9.4 

3 1. 9 1..-Sp .0168 260.549 28.950 3.4 

[ ln (H) J 1 .151 1491.782 1491.782 173.3 

4 ln (CR) 10 .048 479.677 47.967 5.6 

1. 1..-Sp 9 .0158 219.627 23.403 2.7 

[ ~D/D, (~D/D) 2 ] 2, .108 905.914 ~52 ~-95 7 52.6 

Error 878 7556.068 8.606 

21rncrease in explanatory power for regression due to adding variables to the 

model composed of those variables bracketed in the levels above. 

Finite Difference Model 

The dependent variable in this model is ln(H2 /H1). When an expression forb 

analogous to its form in the differential model is inserted in the finite difference 

model (4), we obtain: 

To compare the utility of the two alternative forms of the dependent variable, we 

use the maximum likelihood index of fit (Furnival 1961). The basis for the comparison 

is the standard error of estimate of the untransformed dependent variable. Standard 

errors of estimate for other transformations of the dependent variable are converted 

for comparison by multiplying them by the inverse of the geometric mean of the derivative of the transformation. The derivative of ln (H2 /H1) with respect to ~H is: 

f' (~H) 

'd~H 

1 1 

10 
Accordingly, the inverse of the geometric mean is given by: 

1 

[f' (f1H) ] 

exp [~ ln(H )/n] 

2 

The index-of-fit is 3.54 for the best collection of variables (through level 4) in the 

finite difference model. This index is compared to the standard error of estimate of 

the differential model, which is ±2.93 feet. Hence, the differential model is superior. 

Table 3 is an analysis of variance for the finite difference model with comparisons 

similar to those in table 2. The variables in this model that are analogous to those in 

the differential model are effective as predictors to about the same degree. 

Table 3.--Analysis of improvement of regression model attributable to adding variables. 

Comparison 

level 

1 

2 

3 

4 

5 

Source 

[ ln(D2/D1). ] ~sp 

1. 

~sp 

1. 

~sp 

[ ln(H) J 

[ f1D/D, (f1D/D) 2 ] 

1. 

~sp 

Error 

Dependent variable = ln(H2/H1) 

Proportion of 

: remaining variance 

d.f. : explaine~/ 

Marginal 

sum of squares2_/ 

3.1589 

9 0.091 .3372 

9 .031 .1365 

9 .015 .0754 

1 .252 .7702 

2 .258 .5921 

9 .018 .04633 

886 

He an 

square 

3.1589 

.0375 

.0152 

.00838 

.7702 

.2961 

.00515 

.00185 

F 

1707.5 

20.3 

8.2 

4.5 

416.3 

160.0 

2.8 

~/Increase in explanatory power for regression due to adding variables to the 

model composed of those variables bracketed in the levels above. 

Differential Model-- logarithmic Form 

When the differences between the observed and predicted values of f1H are plotted 

over the predicted values, the variance of the difference increases with larger values 

of the prediction. As a consequence, faster growing trees are given undue weight in 

estimating values of the coefficients. The logarithmic form of the model (3) should 

decrease the variance of the residuals associated with large predictions. The preceding analyses showed that the allometric coefficient (b) varied with species and the 

11 
relation of height, diameter, and diameter increment. These effects could be incorporated quite readily by multiplying each independent variable by a coefficient to be 

estimated, and by introducing a constant term for each species. In the logarithmic 

form, the possibility of zero values for ~D must be considered. Measurements of diameter growth are usually recorded only to the nearest 1/20 inch. By shifting the entire 

scale up 1/20 inch, zero values can be accommodated without distorting the overall 

relation. 

Thus, the logarithmic form of the differential model is: 

ln(~) = c1. + c2 ln(~D + .OS) + c3 ln(D) + c4 ln(H) ~ 

(5) 

For the 909 trees from the northern Idaho forests, the following coefficients for 

this model gave a better index of fit (±2.73 feet) than the differential model with the 

loss of fewer degrees of freedom: 

3.4448 

3.1258 

3.1424 

3.2939 

3.1508 

2.9357 

3.0706 

3.2221 

2.8355 

3.4380 

+0,37401 

-0.29805 

-0.13170 

s2 0.1775 

Western white pine 

Western larch 

Douglas-fir 

Grand fir 

Western hemlock 

Western redcedar 

Lodgepole pine 

Engelmann spruce 

Subalpine fir 

Ponderosa pine 

ln(~D + 0.05) 

ln(D) 

ln(H) 

When the same model was applied to the 265 lodgepole pine and Douglas-fir trees 

from the Lewis and Clark National Forest, the values of the coefficients were quite 

different from those above. The estimated residual variance for these data was 

s 2 = 0.1630. The difference between these two populations is apparent in the following 

averages: 

~H 

~D 

Northern Idaho Forests 

6.5 

1.8 

Lewis and Clark N.F. 

1.4 

0.44 

Differences in growth rates correspond to marked differences in habitat for those 

species that are common to the two geographic sources of data. As a consequence of the 

differing habitats, the Lewis and Clark trees would show a much lower average site index 

than the northern Idaho trees. Furthermore, the Lewis and Clark trees are generally 

older. To accommodate these differences, data from the two sources were merged and the 

differential model modified to permit coefficients to vary with habitat as well as 

species. If all four coefficients were unique with respect to species and habitats, 

then there would be 204 coefficients to be estimated by a separate regression solution 

of equation (5) for each of the 51 entries in table 1. Instead, the four coefficients 

in the logarithmic form of the differential model were varied according to species and 

12 
habitat in repeated regression problems (table 4). The final line represents the best 

model of the series. Coefficients derived from this solution are given in table 5. 

The ratio of the mean-square deviation from regression to the mean-square deviation 

about the mean is 0.2722 for this model. That is, the model accounts for about 73 

percent of the variance of logarithm of height increment among the 1,165 trees in the 

sample. Converting the residual error to the scale of feet of height increment 

(Furnival's index), the mean-square error of estimate would be ±1.97 feet. 

To illustrate the implications of this functional form for height increment in 

the context of stand development, the successive height/diameter curves were plotted 

for a typical even-aged stand of grand fir carried through four decades of simulated 

growth (fig. 4). The curves in figure 4 show that the prediction functions for height 

increment derived in this report can generate successive curves that conform to the 

general shape and level of typical height/diameter curves in even-aged stands shown 

in figure 3. Conformation of these curves for simulated stand development also depends 

on the way diameter increment and mortality change among diameter classes within the 

stand. 

Table 4.--Effect of varying coefficients by species and habitat on the mean square 

residual of the logarithmic form of the differential model 

Number of Variable Mean 

coefficients square 

in model 1 ln(6D+.05) ln (D) ln (H) :residual 

22 H * * H 0:2045 

31 s * H H .2029 

32 S+H * * H .2008 

32 S+H H * * .1958 

40 s H H H .1875 

41 S+H H H * .1853 

Key: * indicates a single coefficient for all species and habitats. 

H indicates a unique coefficient of the indicated variable for each habitat. 

s indicates a unique coefficient of the indicated variable for each species. 

13 
Table 5.--Coefficients for estimating the logarithm of 10-year periodic height growth 

Habitat 

~ N 

~ ~ "-... "-... N tj tj co ;::s. CJ co "-... ~ ~ ~ ~ ~ 

~~ "-...co ·~ co ~ tj tj tj "-...~ ·~ ~ g ~ ~ ~ tj ~ tj co '\jtj ~ tj CJ tj CJ CJ \j;)~ ~$::l.., ~CJ ~ E: tj 

~-~ ·~ E: ·~N ·~ tj ~-~ ·~ ~ ~ 8 ~ ~ tj·~ E: co·~ CON co·~ co ~ co~ ~~ ·~ ~~ tj~ tj;::s. tj co tj·~ ~ CJ ~ ~ ~ tj ~co ~ ~co N CO N~ N ~ N~ CJ~ CJ C) r-g~ ·~ "-...co ·~ ·~ $::l.., ·~ ·~ '\j$::l_, '\j co co~ tj·~ tj~ co~ co C) co C\'2 co ~ ~ ~ ~ ;::s. ~N ~ ~ -~~ \j;)~ ~ ~ ~ ~ ~ ~ ~ ~ ~;::s. ~~ ~ tj ·~ tj ~ ~ ~ tj ·~ tj ·~ ~ ·~ ~ ·~ l:j 

~CD ~~ ~~ ,..Q~ ~l:j ~~ ,..Q~ ,..Q~ ,..Q~ ,_C):::::. ""t! E-!~ ""t! ""t! ""t! ""t! 

Intercept coefficient 

Ponderosa 

pine 3.0970 3.8908 3.4091 

Douglasfir 2.4253 2.7460 2.9966 3.5398 3.0581 3.3143 3.9207 2.4395 2.8073 

Western 

larch 2. 7512 3.5450 3.0633 3.3195 3.9259 2'. 4447 2.8125 

Lodgepole 

pine 2.7973 3.0480 3.5911 3.1094 3.3656 3.9720 2.4908 2.8586 

Grand 

fir 3.6535 3.1718 3.4280 4.0344 

Western 

white pine 3.8157 3.3340 3.5902 2.7154 3.0832 

Western 

redcedar 3.3143 2.8326 3.0888 

Western 

hemlock 3.0256 3.2818 3.8881 

Engelmann 

spruce 2.9080 3.2257 2.3509 1. 6339 

Subalpine 

fir 3.1336 3.7400 2.2588 1.5419 2.6266 

Variable coefficients 

Variable 

1n(6D+0.05) .96586 .11127 .61883 .44920 .27100 .40908 .61661 .05592 .48219 .75458 

ln (D) -.31462 .00278 -.56538 -.35626 -.23160 -.22856 -.47228 .04938 .21472 -.37079 

1n(H) -.1955 -.1955 -.1955 -.1955 -.1955 -.1955 -.1955 -.1955 -.1955 -.1955 

14 
<ll 

~ 

1-

I 

~ w 

I 

120 

110 

100 

_..,...-/~ 

80 

~ 'X 

60 

30 

40 

/ 

20 / 

/ 

DIAMETER (b.h. inches) 

F~gure 4.--Projected height/diameter curves for grand fir on Abies grandis/Pachistima 

habitat. 

Removing Bias in Logarithmic Form 

When the variable to be predicted is analyzed by making a logarithmic transformation, a bias is introduced when the inverse transformation is used to convert the 

estimate to the original natural scale. Effect of the bias would be cumulative when 

these prediction equations are used repetitively to simulate growth of trees and stands 

through successive periods of time. By removing the bias, repetitive application will 

produce the same height from the accumulated predictions of increment as would be the 

result of cumulating observed values of the log-normally distributed height increment. 

Size of the bias depends on the residual variance, which is listed as 0.1853 on 

the last line of table 4, for the logarithmic model. 

Bradu and Mundlak (1970) devised a correction for the bias that recognizes the 

effect of sampling error on the parameters of the regression. Unfortunately, their 

method requires the values of the inverse covariance matrix which is a 41 X 41 array. 

As an alternative, a simple correction that is conditional on the sample values of the 

parameters seems adequate. To determine the corrected estimate of the mean on the 

original scale, the value of one-half the residual variance is added to the estimate on 

the logarithmic scale before making the transformation back to the· natural scale. In 

this case, the amount to be added is 0.1853 t 2 = 0.09265. The effect is to multiply 

the uncorrected estimate of height increment by 1.0971. 

15 
APPLICATIONS 

Calculating Current Annual Increment 

Current annual volume increment is commonly calculated by measuring or estimating 

the two major dimensions of volume-increment: change in tree d.b.h. (inches) and change 

in tree height (feet). Then, these changes are added to present d.b.h. and total 

height, and the volume change is determined from the difference between the volumes 

obtained from a volume table or volume equation that relates tree volume to d.b.h. and 

total height. The prediction equations developed in this report can be used to calculate 

the 10-year periodic height increment from the corresponding 10-year periodic diameter 

increment, initial d.b.h., and initial height. When past periodic diameter increment 

is used to determine future diameter increment, a problem arises because diameter growth 

generally declines with age. Consequently, the current annual rate would differ from 

the periodic rate divided by period length. To remedy this problem, we can invoke the 

observation that whereas diameter increment declines with age, basal area increment 

u~ually remains constant (Spurr 1952, p. 214; Smith 1962, p. 55). 

The sequence of calculations is: 

1. Determine past increase in diameter squared. If increment was measured outside bark, convert to increment inside bark. If measured for a period of y years, 

convert to a 10-year basis. 

DDS (d. 2 - d~ ) * 10/y ~ ~-y 

2. Use past DDS tb estimate future ~D (10-year periodic increment): 

~D = /d. 2 +DDS - d. ~ ~ (inches) 

16 
3. Solve equation (5) using the set of coefficients from the column of table 5 

for the applicable habitat type, and using the first coefficient for the appropriate 

species: 

ln(~H) = e1 + e2 ln(~D + 0.05) + e 3 ln(D.) + e4 ln(H.) '/,- '/,-

~H = exp (ln (~H) + 0. 09265) (feet) 

4. Convert growth estimates to diameter and height 1 year hence; 

ID .z + BKR*DDS/10 

'/,-

where BKR = (D./d.) 2 is the square of the ratio of d.b.h. to diameter inside bark at 

breast height.'L- 'LA computer subroutine prepared in FORTRAN IV to carry out this sequence of calculations is given in the appendix. 

If growth for an arbitrary interval (y) greater than 1 year is desired, then the 

division by 10 in the last two equations would be replaced by multiplication by the 

value of y/10. That is, 

Di+y /Di2 + BKR*DDS*y/10 

Hi+y Hi + ~H*y/10 

17 
LITERATURE CITED 

Bradu, D., andY. Mundlak 

1970. Estimation in log-normal linear models. J. Am. Stat. Assoc. 65:198-211. 

Daubenmire, R., and J. Daubenmire 

1968. Forest vegetation of eastern Washington and northern Idaho. Wash. Agric. 

Exp. Stn. Tech. Bull. 60, 104 p., illus. 

Furnival, George M. 

1961. An index for comparing equations used in constructing volume tables. For. 

Sci. 7(4) :337-341. 

Grosenbaugh, L. R. 

1967. REX--Fortran-4 system for combinatorial screening or conventional analysis 

of multivariate regressions. U.S. For. Serv. Res. Pap. PSW-44, 47 p. 

Perkal, J., and J. Battek 

1955. (An attempt at estimating the development of stands.) (Polish) Sylwan 99(1): 

12-31. 

Smith, David M. 

1962. The practice of silviculture. 578 p. John Wiley, New York. 

Spurr, Stephen H. 

1952. Forest inventory. 476 p. Ronald Press, New York. 

Stage, Albert R. 

1973. A model for the prognosis of forest stand development. USDA For. Serv. 

Res. Pap. INT-137, 32 p. 

Wright, John P. 

1961. Computation of height growth on continuous forest inventory plots. J. For. 

59(10) :772-773. 

18 
APPENDIX 

FORTRAN IV Subroutine 

19 
SUBRQUTINE HTGT(l~P,IhAti,O,H,00,02,H2) 

C I SP IS SPEC It: 5 SUdSCRl PT 

C IHAB IS HABITAT SUBSCRiPT 

C 0 I S 0 I A ME TE R 

C H [ 5 HF IGHT 

C 00 IS lD-YEAR PAST ObH GROWTH 

C 02 IS THE NEW UlAMI:.TER 

C H2 IS THE NE~ HEl~HT 

11 I ME N S I 0 N A S PE ( l u ~ • A ri A ti ( l. 0 ) , B H A B ( 1 0 ) , C H A B ( 10 1 , BK R ( 1 U j 

c 

C SPECIES SUBSCRIPT DEFINITION 

C l ~ -i 1 T l:: PI NE 

C 2 ~l::Sll::RN LARCH 

C 3 DuUGlAS FIR 

C 4 ..,RANO Fl R 

C 5 Wl::STERN HFMLOCK 

C 6 :EDAR 

C 7 LODG~POLF 

C 8 SPRUCE 

C 9 SUbA lt-'1 NE FIR 

C 10 PONUEkOSA PINE 

OATA A SPE /-0.2608.Jl,-O. 531522 ,-Q.536742,-0.423059, 

1 -0.56q264,-0.7ol27l,-o.~s5~39,-0.625377,-0.717423,-0.l857451 

c 

C HABITAT SUBSCRIPT DEFINITION 

C l PSt:UDOTSUGA I PHYSOCARPUS 

C 2 PSlUUOTSUGA I SVMPHORICARPOS 

C 3 P~tUOOTSUGA I CALMAGROSTIS 

C 4 At:Ht:S GRANOIS I PACHIST IMA 

C 5 ABlE~ LASIOCARPA I PACHISTIMA 

C 6 ABltS LASICCARPA I XEROPHY~LUM 

C 1 AcHE:S LASI OCARPA I MENZIES lA 

C 8 ABIES lASICCARPA I VACCINIUM 

C 9 TrtUJA I PACHISTIMA 

C 10 TSUGA HETEROPHYLLA I PACHISTIMA 

DATA AHAB/3.28210,2.96202,2.53339 9 4.07652,4.45741, 

1 2.q7625,2.25931,3.3440o,3.59483,3.85106/ 

DATA BHABI.l1121472,.9o586~1,.61883491,.44919991,.61660659, 

1 .055921353,.4821~198,.75~57698,.27099925,.40907645/ 

DATA CHAB/.00271H9974,-0.314b2085,-0.56538403,-0.35626483, 

1 -0.47227514,.04~37d771,.21472478,-0.37078756,-0.2315979, 

1 -0.22855937/ 

DATA OSP/-O.l955't0131 

DATA BKR /10* 1.09/ 

02 = SQRTCD**2 • OU * C~. * D- ODJ * BKRliSPJ I 10.) 

H2 = H • EXP(ASPE{ISPJ + AHAB(IHAtH + BHAB(IHAB) * ALOG(OO + 0.05) 

1 + CHAB(IHABI * ALU~'D1 • DSP * ALOG(HJ) * 0.109698 

RETURN 

END 

20 

'tr U.S. GOVERNMENT PRINTING OFFICE: 1974 _ 677-092 /30 REGION NO.8 
Prediction of Height Increment 

For Models of Forest Growth 

Albert R. Stage 

USDA Forest Service 

Research Paper INT-164 

Supplement 1 

April 1975 

INTERMOUNTAIN FOREST AND RANGE EXPERIMENT STATION 

Forest Service 

U.S. Department of Agriculture 

Ogden, Utah 84401 

Roger R. Bay, Director 

During the preparation for publication of Research Paper INT-164 

additional trees were obtained from the Colville, Clearwater, and Nezperce 

National Forests, from research studies of grand fir yield by the author, and 

from defect distribution studies by Dr. Arthur D. Partridge, College of 

Forestry, Wildlife and Range Sciences, University of Idaho. 

Analysis of these 330 additional trees has provided better coverage 

of several species and habitats; as shown in the revised table 1. The 

newly derived coefficients for model (5) (pg. 12) are given in the 

revised table 5 and the computer subroutine provided in the appendix. 

The new estimate of the multiplier for removing bias due to the 

logarithmic transformation equals 1.107. 
Table !.--Distribution of sample trees by species and habitat type 

Habitat 

"'-. 

~ "'-. "'-. 8- N "'-. N ~ ~ ~ 

~ 

C) ~ "'-. ~ ~ ~ ~ ~ 

"'-.~ ·~ ~ ~ ~ ~ ~ ~ 

"'-.~ "'-.~ ·~ C) ~ g § ~ ~ 

~ ~ ~ ~ ~ ~ 

~~ ~ N ~ C) ~ C) C) 

t::l)~ a.~ a. a ~ !::: ·~ !::: ·~N ·~ ~ ·~ § ~·~ 

~ ~ ~ N ~·~ ·~ ~·~ ~·~ ~N ~·~ ~ N ~ a. N~ ~ ~~ ~~ ~ ~ ~ ~ ~·~ 

~C) ~ ~ ~ ~ a.~ ~ ~~ N~ N~ N ~ N ~ 

C)~ 

~ ~ ~~ ·~ "'-.·~ ·~ ·~ ~ ·~ ·~ 

'""(j~ ~~ ~~ ~~ ~~ ~ C) ~ t\2 ~ ~ 

~ !::: ~ ~ ~N ~ ~ •o;--;, ~ a.~ ~ ~ ~ N ~ ~ ~ ~ 

~~ ~~ ~ ~ ·~ ~ ~~ ~~ ·~ (j ·~ ~ ·~ ~ ·~ ~ 

~tr::l ~~ ~u ,..(:)~ ,..(:)~ ,..(:)~ ,..(:)~ ,..(:)::::;;: 

SJ2ecies ~ ~ ~ "!! E-t E-t "!! "!! "!! "!! Total 

Ponderosa 

pine 1 30 17 8 56 

Douglas-fir 108 107 28 143 31 41 27 6 6 497 

Western larch 9 17 14 27 4 14 2 87 

Lodgepole 

pine 7 49 14 3 17 3 12 4 90 199 

Grand fir 81 48 89 17 3 1 239 

Western white 

pine 22 3 76 1 1 103 

Western 

redcedar 1 52 39 92 

Western 

hemlock 3 109 8 120 

Engelmann 

spruce 2 2 3 3 6 7 23 

Subalpine fir 9 16 42 11 1 79 

Total 109 153 79 297 162 410 78 84 22 101 1,495 
Table 5.--Coefficients foP estimating the logaPithm of 10-yeaP pePiodic height gPowth 

Habitat 

" ~ 

" N " " .......... N ~ ~ ~ ~ Cf,) ~ 

~ ~ ~ ~ a Cf,) .......... ~ ~ 

.......... ~ ·~ Cf,) ~ ~ ~ ~ ~ '-.._Cf,) "~ ·~ 2 ~ It> g ~ It> It> ~ ~ ~ ~ ~ Cf,) '\j~ ~ a ~ a a \:).)It) 

~~ ~2 ~ E: E: ~ E: ·~ E: ·~N ·~ ~ ·~ 

Cf,) 

~·~ 

~ Cf,) ~ Cf,) \:).) ~·~ ·~ ~·~ Cf,)·~ Cf,)N Cf,)·~ Cf,) ~ ~~ ~ ~~ ~~ ~ ~ ~ Cf,) ~ ·~ ~ a ~ It> ~ ~ \:).)Cf,) Cf,) ~Cf,) N Cf,) N~ N~ N ~ a~ 

~ ~ ~8 ·~ '-.._·~ ·~ ·~ ~ ·~ ·~ '\j~ Cf,)~ ~~ ~~ Cf,)~ Cf,) a Cf,) t\1 Cf,) It> ~ E: ~ ~ ~N ~ It> '\""";>It> \:).)It) ~ It> ~ ~ ~ ~ ~ It> ~ ~ ~~ ~ ~ ·~ ~ 

~~ ~ ~ ·~ ~ ·~ ~ ·~ ~ ·~ 

~CQ ~~ Cf,)Cj ,..Q~ ~~ ...Qr::l:t ,..Q~ ,..Q~ ,..Q ~ ~ "'~:! "'~:! ~ ~ "'~:! 

IntePcept coefficient 

Ponderosa 

pine 2.4768 3.9227 3.4004 2.6014 

Douglasfir 2.1560 3.6019 2.8472 3.0796 2.2806 2.9937 3.0606 1.9315 2.4721 

Western 

larch 3.6605 3.1382 2.3392 3.0523 3.1192 1. 9901 -. 2. 5307 

Lodgepole 

pine 3.5812 2.8264 3.0589 2.2599 2.9730 3.0399 1. 9108 1.7232 2.4513 

Grand 

fir 3.2032 2.4042 3.1172 3.1842 2.0550 2.5956 

Western 

white pine 3.3438 2.5448 3.2579 2.1957 2.7363 

Western 

redcedar 2.7946 1.9956 2.7086 

Western 

hemlock 2.2455 2.9586 3.0255 

Engelmann 

spruce 2.7679 3.0003 2.9144 2.9813 1.8522 1.6646 

Subalpine 

fir 2.9458 3.0128 1.8836 1.6960 2.4242 

VaPiable coefficients 

Variable 

ln(6D+0.5) .99173 .56944 .63363 .46651 .49104 .43979 .50854 .33112 .95970 .79813 

ln(D) -.35739 -.62543 -.64966 -.34633 -.09599 -.27165 -.36298 .05146 .02819 -.35941 

ln(H) -.09613 -.09613 -.09613 -.09613 -.09613 -.09613 -.09613 -.09613 -.09613 -.09613 
SUB R 0 I JT I N F H T G T ( I S P , I H A R , 0 , H, 0 0 , 0 2 , H 2 ) 

c 

C ISP IS SPECIES SUBSCRIPT 

C IHAB IS HABITAT SUBSCRIPT 

C D IS f)JAMETER 

C H IS HFIGHT 

C D1J IS 10-YEAR. PAST DBH GROWTH 

C 02 IS NF~ DIAMETER ONE YEAR HENCE 

C H2 IS THE NE~ HEIGHT ONE YEAR HENCE 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

r 

c 

c 

c 

c 

r 

c 

( 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

c 

'( 

c 

c 

OIMENSICN ASPE(lO),AHAB{lO),BHAB(lO),CHAB(lQ),BKk{lO) 

SP f C I E S SUR SC R I P T DE F I N I T I C t\ 

l WHITE P[NE 

2 ~ESTERN LARCH 

3 DOUGLAS FIR 

4 GRANO FIR 

5 WESTERN HEMLOCK 

6 CEDAR 

7 LCOGEPOLE 

8 SPRUCE 

9 SUB4LPINE FIR 

lO PCNOEROSA DINE 

DATA ASP~ I-0.7319SO,-C.937502,-0.996l32,-0.R72584,-l.03127, 

1 -1.23118, -1.01687, -1.07542, -1.04398, -0.6?5301 

1 

l 

1 

2 

HAbITAT SUtj SCRIPT DEFINITIOt\ 

DATA AHAB 

DATA f3HA3 

DATA CHAB 

l PSEUOOTSUGA I PHYSIJCARPU5 

2 PSEUDOTSUGA I SYMPHORICARPOS 

3 PSEUDOTSUGA I CALMAGRCSTIS 

4 ,1.B I ES GP/\"JDtS I P/\CHIST IMA 

5 ABIES LASICCAR.PA I PACHISTIMh 

6 ABIES LASIOCARPA I XEROPHYLLUM 

7 ARIES LASICC.\P.PA I MENZIESIA 

8 ABIES LAS l CCl\~ PA I V/\CCINIUM 

C) THUJA I PACH IS T I M.A 

10 TSUGA HETERCPHYLLA I PACHISTIMA 

14.59802,3.15211,3.84330,4.07574,4.05676, 

2.92763,2.74002,3.46821,3.27674,3.989831 

I • 56 94 3 6 7 9 , • 9 9 1 7 3 ll 1 , • 6 3 3 6 2 7 9 5 , • 4 b 6 5 1 0 12 , • 50 8 54 2 0 C, 

.331119~5,.95970452,.7Q8127ll,.49l038~0,.4397864gl 

/-0.62542897,-0.35739297,-0.64966339,-0.34632784, 

-0.36297816,.05l4b3l76,.02Rl94819.-.35940510, 

-.095985949,-0.271645071 

8ATA DSP 1-0.09612711 

D2 = SQRT(D>:{*2 +DO* (2. * 0- DO)* BKR( ISP) I 10.) 

H 2 = H + E X P ( A SP E { I S P ) + A H A B ( I H A B ) + B H A B ( l H A B ) * A L 0 G ( 0 0 + 0 • 0 5 ) 

l + CHAR(IHAB) * ALCG(OJ + OSP * ALOG{H)) * 0.11066 

FE TURN 

END 

";:r U.S. GOVERNMENT PRINTING OFFICE: 1974 _677-696/55 REGION NO.8 
ALBERT R. STAGE 

1975. Prediction of height increment for models of forest growth. USDA 

For. Serv. Res. Pap. INT-164, 20 p., illus. (Intermountain Forest 

and Range Experiment Station, Ogden, Utah 84401.) 

Functional forms of equations were derived for predicting 10-year periodic 

height increment of forest trees from height, diameter, diameter increment, 

and habitat type. Crown ratio was considered as an additional variable for 

prediction, but its contribution was negligible. Coefficients of the function 

were estimated for 1G species of trees growing in 10 habitat types of northern 

Idaho and northwestern Montana. 

OXFORD: 561.1; 564. KEYWORDS: height increment, tree-growth models, 

western white pine, western larch, Douglas-fir, grand fir, western hemlock, 

western redcedar, lodgepole pine, Engelmann spruce, subalpinefir, ponderosa 

pine, habitat type. 

ALBERT R. STAGE 

1975. Prediction of height increment for models of forest growth. USDA 

For. Serv. Res. Pap. INT-164, 20 p., illus. (Intermountain Forest 

and Range Experiment Station, Ogden, Utah 84401.) 

Functional forms of equations were derived for predicting 10-year periodic 

height increment of forest trees from height, diameter, diameter increment, 

and habitat type. Crown ratio was considered as an additional variable for 

prediction, but its contribution was negligible. Coefficients of the function 

were estimated for 10 species of trees growing in 10 habitat types of northern 

Idaho and northwestern Montana. 

OXFORD: 561.1; 564. KEYWORDS: height increment, tree-growth models, 

western white pine, western larch, Douglas-fir, grand fir, western hemlock, 

western redcedar, lodgepole pine, Engelmann spruce, subalpinefir, ponderosa 

pine, habitat type. 

ALBERT R. STAGE 

1975. Prediction of height increment fo

For. Serv. Res. Pap. INT-164, 

and Range Experiment Station, O

Functional forms of equations were deriv

height increment of forest trees from heigh

and habitat type. Crown ratio was conside

prediction, but its contribution was neglig

were estimated for 10 species of trees grow

Idaho and northwestern Montana. 

OXFORD: 561.1; 564. KEYWORDS: height 

western white pine, western larch, Douglas

western redcedar, lodgepole pine, Engelman

pine, habitat type. 

ALBERT R. STAGE 

1975. Prediction of height increment fo

For. Serv. Res. Pap. INT-164, 

and Range Experiment Station, O

Functional forms of equations were deriv

height increment of forest trees from heigh

and habitat type. Crown ratio was conside

prediction, but its contribution was neglig

were estimated for 10 species of trees grow

Idaho and northwestern Montana. 

OXFORD: 561.1; 564. KEYWORDS: height 

western white pine, western larch, Douglas

western redcedar, lodgepole pine, Engelman

pine, habitat type. 
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)