United Slates 

Department of 

Agriculture 

Forest Service 

Northeastern Forest 

Experiment Station 

Research 

Paper N E-493 

or Even-Aged 

Oak Stands 

by Donald E. H i l t 

and Martin E. Dale 
The Authors Abstract 

Donald E. Hilt received his 

bachelor's degree in forestry from 

lowa State University in 1969, and a 

master of science degree in forestry 

from Oregon State University in 

1975. He joined the Northeastern 

Forest Experiment Station in 1975 

and worked as a statistician in the 

biometries group at Upper Darby, 

Pennsylvania. In 1976 he transferred 

to Delaware, Ohio, where he is a research forester specializing in studies of growth and yield of managed 

upland oaks. 

Martin E. Dale joined the Forest 

Service in 1957 and specialized in 

forest management and silviculture 

research at Berea, Kentucky. In 

1973, he received a Ph. D. degree in 

forest biometry from lowa State 

University, and since i970 has been 

assigned to the Northeastern Forest 

Experiment Station's research unit 

on timber measurement and management planning at Delaware, 

Ohio. Currently he is specializing in 

studies of growth and yield of managed upland oaks. 

Forest growth models that use 

predicted tree diameters or diameter 

distributions require a reliable 

height-prediction model to obtain 

volume estimates because future 

height-diameter relationships will 

not necessarily be the same as the 

present height-diameter relationship. A total tree height prediction 

equation for even-aged upland oak 

stands is presented. Predicted tree 

heights follow a biologically consistent progression with increasing 

dbh, age, and site index. The consistent progression of heights does 

not allow erratic or illogical volume 

increments. The proposed equation 

satisfactorily predicted the heights 

of 4,619 oak trees measured in six 

states. 

Manuscript received for publication 

15 June 1981 
Diameter-growth models for 

individual trees, diameter distribution models, and stand table projection methods are all examples of 

forest growth modeling techniques 

that use predicted tree diameters or 

diameter distributions to estimate 

future stand characteristics. These 

types of forest growth models are 

receiving increased attention because they provide detailed information on the structure, and in some 

instances the species composition, 

of the future stand. Since the 

volume of the future stand is also of 

interest, heights must be assigned 

to predicted tree diameters or to the 

diameter classes of a predicted 

diameter distribution to obtain 

volume. 

Methods for estimating future 

heights should be thoroughly investigated because future heightdiameter relationships will not 

necessarily be the same as the present height-diameter relationship 

(Chapman and Meyer 1949). The preferred method of assigning future 

heights is to model height growth 

as a function of variables such as 

tailed stem analyses of felled trees, 

An alternate method for estimating 

future heights is to construct a general model that expresses total free 

height as a function of variables 

such as dbh, age, and site index. 

This paper presents a method 

for predicting total tree heights that 

follow a biologically consistent progression with increasing dbh, age, 

and site index. The consistent progression of heights does not allow 

erratic or illogical volume increments. The resulting height prediction equations are applicable to 

even-aged upland oak stands. 

Data 

Data for developing and testing 

the height equations were taken 

over a range of age and site conditions in unmanaged even-aged upland oak stands in Ohio, Kentucky, 

Missouri, Iowa, Illinois, and Indiana. 

A total of 2,306 felled-tree heights 

on 150 plots (stands) from an oak 

decay study were used in the analysis. Plot size was 0.08 ha (115-acre). 

These data were augmented with 

2,313 standing-tree heights measured with a Spiegel-relaskopl on 158 

0.2- and 0.4-ha (112- and I-acre) 

growth and yield plots. Bbh and total height were measured on trees 

from all crown classes in both 

studies. Total age was determined 

by counting annual rings for felled 

trees, and increment borings for 

standing trees. Site index, the 

height attained by the average dominant and codominant oak at total 

age 50, was determined for each 

plot from Schnur's (1937) site index 

curves for upland oaks. 

The data were divided for analysis according to species with similar growth patterns. The black oak 

group of 2,116 heights consisted of 

black oak (Quercus velutina Lam .), 

scarlet oak (Q. coccinea Muenchh.), 

and northern red oak (Q. rubra L.). 

The white oak group of 2,503 

heights consisted of white oak (Q. 

alba L.) and chestnut oak (Q. prinus 

L.). Site index for both species 

groups ranged from approximately 

50 to 80, age from 25 to 125 years, 

and dbh from 6.6 to 66 crn (2.6 to 26 

inches). 

Analysis 

Mean black oak heights of varidbh, age, and site index. However, ous dbh x age x site index cate- ' The use of trade, firm, or corpora- data necessary the gories plotted in Figure 1 show that tion names in this publication is for the of a height-growth model are not information and convenience of the dbh, age, and site index are imporavailable at the present time for the reader. Such use does not constitute an tant factors related to tree height. 

upland oak timber type. Reliable official endorsement or approval by the Similar relationships were observed 

height growth data for the U.S. Department of Agriculture or the 

Forest Service of any product or service on other sites for black oak, and 

deliquescent-branching upland oaks to the exclusion of others that may be also for the white oak data. Our goal 

can only be obtained through de- suitable. was to build the observed relationFigure 1. Average heights for black oak by dbh, age, and site index. 

TOTAL HEIGHT (METERS) 

SITE INDEX 60 

TOTAL HEIGHT !METERS) 

SITE INDEX 70 

25 

20 

30 --- 15 40 b 

50 0 

60 

70 - 10 80 0 

90 + 

100 x 

5 

5 I0 15 20 25 30 35 40 45 50 5 10 15 20 25 30 35 40 45 50 

Dbh icmJ Dbh (cm) 
ships into the height prediction 

model. The following conditions for 

a logical and consistent height prediction equation were proposed: 

I, Height equals breast height 

(I -37 rnj when dbh equals 

zero. 

2. Height increases at a decreasing rate as dbh increases. 

3, Height increases at a decreasing rate for a given dbh 

as age increases. 

4. Height increases for a given 

dbh and age as site index increases. 

Linear regression models for 

heig ht-d iarneter-age equations were 

explored by Curtis (1967). Many of 

these proposed models, including 

site index, were fitted to the oak 

data. While many of the fitted equations produced R2 values near 0.80, 

none fulfilled all of the above conditions. Some equations "peaked" 

within the data, predicting shorter 

heights for larger diameters. Other 

equations failed to produce reasonable maximum heights for a given 

age and site combination. 

Meyer (1940) investigated a 

modified exponential heig ht-diameter model of the form 

H = 1.37 + 0 [I -exp (-J3D)], 

where t-i = total height in meters, D 

= dbh in centimeters, and exp is 

the base of the natural logarithms. 

The intercept of this model is 1.37 

m (breast height). Height increases 

at a decreasing rate as dbh increases, but can not exceed the 

asymptotic height, a + "I37 meters. 

The parameter p determines the rate 

at which the curve approaches the 

asymptotic height. Meyer Found that 

this model provides an excellent fit 

to observed data for trees larger 

than about 5 inches dbh and older 

than 10 to 20 years. For practical 

purposes then, this curve is satisfactory for height-diameter equations For a given age and site index. 

It fulfills conditions 'I and 2 stated 

previously. 

Meyer's equation works well for 

a specific stand at a given age and 

site index because a and 13 are 

usually estimated with a sample of 

dbh, and total height data. To obtain 

a general model that is suitable 

when sample heights are not available, we needed a method based on 

stand age and site index for estimating the parameters a and p. 

The proposed height prediction 

model has the form 

H,, = 1.37 + a, [I - exp 

( - 01 Dill], (1 1 

where H,, = height of the jth 

tree in the ith stand, 

D,, = dbh of the jth tree 

in the ith stand, 

a, = asymptotic height 

parameter for the ith 

stand, 

and 13, =. slope parameter for 

the ith stand. 

In this form the model is not useful 

as a height prediction model because it is stand-specific. One way 

to build a general height prediction 

equation would be to model the 

parameters a, and J3, as functions of 

stand age and site index. The 

parameters of such a stochastic coefficients model could be estimated 

by the two-stage procedure proposed by Ferguson and Leech 

(1978). However, the two-stage least 

squares computer program for estimating the coefficients is not 

readiiy available. The approach we 

use in modeling the parameters is 

based on existing site index curves 

and stand tables for normal stands. 

We First assume that the 

asymptotic height parameter, Q,, For 

any stand is related lo the maximum 

tree height, and the maximum tree 

height is related to average height 

of the dominant and codominant 

trees in a stand (height of the site 

trees). Based on experience and investigation of the data, the maximum height is assumed to be a csnstant percentage "ratler than the 

mean height of the site trees. Since 

"ie asymptotic height for a given 

stand is a, + "1.37 meters: we can 

express a, as a function of "re mean 

height of the site trees for the 

stand, H,,: 

a, + 137 = kKir 

or a, = kHs, - 1.37, 

where k is greater than one. 

(2) 

We next condition the height 

equati-on to pass through the point 

(D,,, H,,) for a given stand, where D, 

is the mean dbh of the site trees in 

the im stand. The condition that has 

to be met is 

Solving for Pi gives 

J3, = - 1 n [(aj + 1.37 

- Plsrjt*i] &,- '. (3) 

Substituting the value of from (2) 

into (3fl 

= - 1 n [(k Es, - Rsi)/ 

Substituting the right hand sides of 

equations (2) and (4) into equation 

(1) gives a conditioned height model 

which is written as 

Hij = 1.37 + (k &, - 1.37) x 

{I - exp [ln[(k KS, - KSi)l 

( - 137 1- D] (5) 

The conditioned height model 

is nonlinear with one unknown 

parameter, k. The model is still 

stand-specific and not useful as a 

height prediction equation unless 

the mean height and dbh of the site 

trees can be modeled as functions 

of stand age and site index. 

We used values read from 

Schnur's site index curvesto model 

the relationship between H,: stand 

age and site index. The mean height 

of the site trees, regardless of oak 

species, can be read directly from 

Schnur's curves for any age and site 

index. We filled Schnurk curves to a 

modified Richards' growth function 

(Ek 1971 and Payandek 19774. The 

mean heigli"rf the site trees can be 

estimated as a function of stand age 

(A) and site index IS) with the following equation: 

Ks = 1.80408 S932m"'l - exp 

( - .0240308 A)j2 55529S- 280445 

(6) 
Figure 2.. Nonlinear equations for Schnur's site index curves. Circles represent points read from Schnur's curves. 

H, TOTA t HEIGHT (METERS) f H~= 1 80408 s ,932097 (1-,-.0240308 A) 55529 S- 280445 

SITE INDEX 

METERS ( FEET) 

10 20 30 40 50 60 70 80 90 iOO 110 120 

AGE (YEARS) 

The equation fitted the site curve 

data very well (Fig. 2). Ths R2 was 

greater than 0.99. Since H, is a function of stand age, eguation (6) can 

be used to project H, to some future age. 

If data are collected from a 

large number of stands, observed 

values of D, could be used to 

model the relationship between D, 

stand age, and site index. Although 

we had a large amount sf data, the 

trees used to determine the site index of a plot could not be identified. 

To circumvent this problem, we 

used the mean diameter of the trees 

in the upper 20 percent of the 

diameter distribution from Schnur's 

stand tables as our estimate of D, 

for a given stand. Our experience in 

upland oaks has shown that the site 

index trees are generally in the upper 20 percent of the diameter distribution for the trees in a stand, A 

field test of this assumption on 32 

permanent growth and yield plots 

revealed that on the average 75 percent of the site trees selected on a 

given plot were in the upper 20 percent of the diameter-distribution for 

that plot. Values of D, calculated 

from Sc hnur's stand tables increase 

with increasing stand age and site 
index (Figs. 3a-3b). We used a modified Richards' function to model the 

relationship between the mean 

diameter of the site trees, stand age 

(A), and site index (Sj. The resulting 

equation for black oaks, 

- 

D, = 5.49927 SX4""' " - e~p 

(- ,0192593 A)]' 25342, 

had an R2 greater than 0.99. (7) 

White oaks are generally 

smaller in diameter than black oaks 

for a given age and site index. 

Therefore, we fitted separate curves 

for white oaks based on Schnur's 

stand tables, The resulting equation 

for white oak, - 

Ds = 6.40146 S.637myCI - exp 

1- ,022761 4 (8) 

also had an R2 greater than 0.99. 

.-* 

we caicuiated vaiues of hS, and 

&, for each stand with equations 

(6), (7), and (8). These values, along 

with the observed values of t-l,, and 

D,, were used to estimate the value 

of the parameter k in the conditioned height model, equation (5). 

Our initial estimate ofthe 

asymptotic height (k ti,) for trees at 

a given stand age and site index, 

based on experience and investigation of the data, was 10 percent 

greater than the average height of 

the site trees. That is, K = 1.1. Our 

final estimate of k was determined 

with an iterative fitting routine that 

minimized the percentage differences between estimated and actual 

heights (estimated height less actual height x 1001 actual height). 

Results 

Black oaks. A k value of 1.07 resulted in a model that fitted the 

black oak data well. The average of 

the percentage differences for the 

21 16 black oak trees was - 0.15 percent with a standard deviation of 

9.05 percent, and the calculated R2 

was 0.79. The fitted curves are 

shown in Figures 4a-4c. 

Figure 3a. Average diameters of the largest 20 percent: of the trees for 

black oak, by age and site tndex. C~rcles represent points calcutated from 

Schnur's stand tables. 

Ds,AVERAGE DBW OF LARGEST 20% OF TREES iCENTlMETERSf 

21 34 (701 

SITE INDEX '' 29'60' METERS (FEET) 

15 24 (501 

iC 20 30 4C 50 60 70 8G 90 ICO 110 120 

AGE (YEARS) 

Figure 3b. Average diameters of the largest 20 percent of the trees for 

white oak, by age and site index. 

Ds, AVERAGE DBH OF LARGEST 20°/0 OF TREES (CENTIMETERS) 

SITE INDEX 

METERS (FEET) 

The average percentage differences were then tabulated by age 

and site index categories to check 

for bias in certain parts of the curve 

AGE ( YEARS) 
Figure 4a. Height-diameter-age curves for ToTaL HEIGHT (METERS) 

black oak, site indexJ5.3 m (50 feet). Circles 

represent the point (Ds, Hsf that curve is I 

forced through. i 25 kI 

Figure 4b. Black oak site index 18.29 rn (60 feet). TOTAL HEIGHT (METERS) 

I I 1 I I I I I I I 

li 10 15 20 25 30 35 40 45 50 

Dbh (cml 

Figure 4c. Black oak site index 21.34 rn (70 feet). TOTAL HEIGHT (METERS1 

30 r 

I L 1 I I I 8 I 1 I I 

5 I0 15 20 25 30 35 40 4 5 50 

Dbh (cml 
(Tabte 1). Most of the percentage 

differences, especially for those age 

x site index categories with a large 

number of trees in them, were within +- 4 percent. There was a slight 

trend from negative to positive bias 

with increasing age and site index. 

The percentage differences 

were also tabulated by age, site index, and dbh as illustrated in Table 

2 for site index 70. Except for the 

slight trend with age, no significant 

bias was detectable. Differences for 

other site index categories were 

similar. 

White oak. The value of k for 

white oak was 1.12. The resulting 

average percentage difference for 

the 2503 white oak trees was 0.42 

percent with a standard deviation of 

10.0 percent, and the calculated R2 

was 0.81. The magnitude and trends 

of the percentage differences were 

similar to those for black oak. The 

fitted curves are shown in Figures 

4d-4f. 

We feel that the general height 

model does a good job of predicting 

individual tree heights. Conditioning 

the heightmodel to pass through 

the point (D,, H,) guarantees logical 

height predictions for a given stand. 

All of the conditions stated previously for a logical and consistent 

height prediction equation were 

met. 

Table 1 .-Average percentage differences in height 

by age x site index categories for black oak. 

Site index class 

15.24 m 18.29 m 21 '34 m 24.38 m 

Age (53 feet) (60 feet) (70 feet) (80 feet) 

No. of Average % No. of Average Oh No. of Average % No. of Average % 

trees deviation trees deviation trees deviation trees deviation 

' Percentage difference = [(estimated height-actual height)lactual height] x 100. Only those age x site categories with five or more trees are shown. 

Table 2.-Average percentage differences of age x dbh categories 

for black oak, site index 21.34 m (70 feet).' 

Dbh Glass 

Age l0.16cm 15.24cm 20.32cm 25.40cm 30.48cm 35.56cm 40.64cm 45.72cm 

(4 inches) (6 inches) (8 inches) (10 inches) (12 inches) (14 inches) (16 inches) (18 inches) 

Tercentage difference = [(estimated height-actual height)lactual height)] x 100. Only those age x dbh categories with five or more 

trees are shown. 

6 
" Figure 4d. Weight-diameter-age curves for TOTAL HEIGHT (METERS) 

white oak, site index-15.3 rn (50 feet). Circles 30 

represent the point (Ds, Ws) that curve is 

forced through. 

I I i I I I I i I 1 i 

5 10 15 20 2 5 30 35 40 4 5 50 

Dbh (cm) 

Figure 4e. White oak site index 18.29 rn (60 feet). TOTAL HEIGHT (METERS) 

Figure 4f. Whiteoak site index 21.34 rn (70 feel). TOTAL HEIGHT( METERS) 
Thinned Stands 

The proposed height equation 

predicts heights of trees from unmanaged sZands very well. However, 

many growth simulation models are 

developed from managed stands. Increased diameter growth of residual 

trees after thinning would necessarily alter the height-diameter relationship at future ages if height 

growth is not affected. However, the 

effect of thinning on height growth 

of residual trees may neutralize any 

change caused by increased diameter growth, or it may even augment 

the change. Changes in the heightdiameter relationship due to thinning are beyond the scope of this 

paper. Our intention here is to determine whether the height equations 

developed in this paper for unmanaged stands perform satisfactorily 

for thinned stands as well. 

Sample tree heights were measured in 1977 on four different thinning studies, 16 years after thinning. 

The thinning method used is best 

described as "freeH-the marker 

was free to remove trees from all 

crown classes. The degree of thinning was controlled by reducing the 

basal area or the stocking level to 

the desired percentage. Stocking 

percent (Gingrich 1967) less than 50 

represents a heavily thinned stand: 

50-75 percent a medium thinning, 

and 75+ percent a light thinning. 

Sample tree heights were predicted 

by using the height models developed in this paper, and the average percentage differences were 

tabulated by study and stocking 

level (Table 3). 

The height equations predicted 

the sample tree heights satisfactorily. Even though some of the 

study x stocking level categories 

had a limited number of trees, none 

of the average percentage differences exceeded 10 percent. Although not conclusive, these results 

indicate that thinning did not dramatically alter the height-diameter 

relationship present in unthinned 

stands. The effects of thinning on 

the height-diameter relationship 

may become apparent when thinned 

stands are observed after a period 

longer than 16 years. 

Tree heights estimated by the 

method presented in this paper follow a logical and consistent progression with increasing tree dbh, 

age, and site index. The conditions 

imposed on the height equations 

prevent erratic and illogical treeheight predictions that sometimes 

occur with traditional regression 

techniques. The equations can be 

inserted with only a few programming statements into many forestgrowth computer routines. 

The height equations are intended for use with growth models 

for upland oaks that involve the prediction of tree diameters or diameter distributions to estimate future 

stand characteristics. We do not 

advocate using these height equations for existing stands of known 

age and site index, Height-diameter 

equations should be constructed for 

such stands from sample tree 

heights and diameters. 

While we have demonstrated 

that the height equations perform 

satisfactorily for thinned stands, 

changes in the height-diameter relationship due to thinning need to be 

investigated more thoroughly in future studies. 

Table 3. Average percentage differences for stands 16 years after thinning.' 

Initial Avg. Thinning Predominant avg. age site 

study species (years) index 

1 White oak 34 70 

2 Black, 

scarlet oak 34 73 

3 Black, 

scarlet oak 62 64 

4 White oak 80 64 

Stocking level (percent) 

<50 50- 75 75 + 

No. of Average % No. of Average % No. of Average % 

trees deviation trees deviation trees deviation 

56 - 3.4 38 - 9.5 38 - 3.3 

Percentage difference = [(estimated height-actual heightjiactual height)] x 100. Only those age x stocking categories with five or 

more trees are shown. 

8 
literature Cited 

Chapman, H. H., and W. H. Meyer 

1949. Forest mensuration. 1 st 

ed. McGraw-Hill, New York, 

522p, illus. 

Curtis, R. 0. 

1967. Height-diameter and 

he ight-diameter-age equations 

for second growth Douglas-fir. 

For. Sci. 13:365-375. 

Ek, A. R. 

1971. A formula for white spruce 

site index curves. Univ. Wis. 

For. Res. Note 161. 2p. 

Ferguson, I. S. and J. W. Leech. 

1978. Generalized least squares 

estimation of yield functions. 

For. Sci. 2427-42. 

Gingrich, S. F. 

1967. Measuring and evaluating 

stocking and stand density in 

upland hardwood forests in the 

Central States. For. Sci. 

13:38-52. 

Meyer, A. H. 

1940. A mathematical expression for height curves. J. For. 

38:415-420. 

Payandeh, B. 

1974. Formulated site index 

curves for major timber species 

in Ontario. For. Sci. 20:143-144. 

Schnur, L. G. 

1937. Yield, stand, and volume 

tables for even-aged upland oak 

forests. U.S. Dep. Agric. Tech. 

Bull. 560. 87p. 

Hilt, Donald E. and Martin E. Dale. Height prediction equations 

for even-aged upland oak stands. Broomall, Pa.: Northeast. 

For. Exp. Stn.; 1982; USDA For. Serv. Res. Pap. NE-493,9p. 

A total tree height prediction equation for even-aged upland oak 

stands is presented. Predicted tree heights follow a logical and 

consistent progression with increasing dbh, age, and site index. 

ODC 522.2,522.31,561 .l 

Keywords: Height prediction; upland oaks 

U . S . GOVEiUi?iE?S:T TRIhTi3G OFFICE : 1982-505-0 i 2 : 3 
Headquarters of the Northeastern Forest Experiment Station are in 

Broomall, Pa. Field laboratories are maintained at: 

@ Amherst, Massachusetts, in cooperation with the liniversity of 

Massachustts. 

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@ Burlindon, Vermont, in cooprration with the University of 

Vermont, 

@ Delaware, Ohio, 

@ Durham, Vew Hampshire, in coopration with the k niversitj of 

New Hampshire, 

@ Hamden, Connecticut, in coopration with Yale lirriversit~. 

@ Morgantowra, West 'k'irginia, in coopration with West Virginia 

University, hiflorgantown. 

@ Orono, kilaine, in cootperation with the University of Maine, 

Orons. 

@ Parsons, West Virginia, 

@ Princeton, West Virginia. 

@ Syracrrw, New Uork, in coopsation wilh tkae State Ijrriversity csf 

New Uork Corilege of Environmental Sciences and Forestry al 

Syracuse University, Sj racuw- 

@ bniversity Park, Pennsplkania, in coopration N ith the 

Pennsylvania State University. 

@ Warren, Pennsj lvania.