United States 

Department of 

Agriculture 

Remarks on Height-Diameter 

Modeling 

Forest Service 

Lei Yuancai and Bernard R. Parresol' 

Southern 

Research Station 

Research Note 

SRS-10 November 2001 

Abstract 

Height-diameter model forms in earlier published papers are examined. 

The selection criteria used in height-diameter model forms are not 

reasonable when considering tree biological growth pattern. During model 

selection, forms for height-diameter relationships should include 

consideration of both data-related and reasonable biological criteria, not 

just data-related criteria. A reasonable model form should possess the Sshaped Functional roperties of monotonic increment (dyldt > O), 

inflection point (d S y/dt2 = 01, and asymptotical value b+asymptote as 

t-w), rather than the concave shaped, functional properties. The Sshaped models are best when appropriate data are collected (data sets 

contain~ng early growth). The Bertalanf@-Richards and Schnute functions 

are recommended for modeling height-diameter relationships, and an 

example contrasts the fit of the sigmoidal Schnute function against the 

concave Meyer function. 

Keywords: Height-diameter, model form, Schnute function, selection 

criteria, sigmoid growth. 

Introduction 

Foresters often use height-diameter models to predict total 

tree height (1F-I) based on observed diameter at breast height 

(d.b.h.) for estimating tree or stand volume and site quality, 

e.g., Stout and Shumway 1982. Therefore, estimations of 

tree or stand volume and site quality rely heavily on 

accurate height-diameter functions. There have been many 

postulated height-diameter models for different species and 

regions since Meyer (1940) suggested the exponential 

height-diameter relationship: 

i 

Staff Member. Research Institute of Forest Resource Infomatton 

Techniques, Chinese Academy of Forestry, Beljlng 100091, People's 

Republic of China; Mathematical Statistician, U.S. Department of 

Agriculture, Forest Service, Southern Research Station, Southern Forest 

Inventory, Monitoring, and Analysis Program, Ashev~lle, NC 28804, 

respectively. 

where 

k - breast height (4.5 feet (ft) [1.37 meters (m)] 

aboveground level in the United States, 1.3 m (4.3 ft) in 

countries using the metric system) ; and 

bi = a coefficient. 

Sigmoid and Concave Curves: Competing Functions 

Huang and others (1992) selected and compared 20 

nonlinear height-diameter functions, fitted by weighted, 

nonlinear least-squares regression for white spruce (Picea 

glauca Voss.) and aspen (Populus tremuloides Michx.). 

Their selection was based on examination of heightdiameter relationships as revealed by plotting H against 

DBH for the two species. Based on their examinations, 

Huang and others (1 992) believed that the height-diameter 

relationship for white spruce was a typical sigmoid or Sshaped curve with an inflection point occurring in the lower 

portion of the data points (fig. 1). The height-diameter 

relationship for aspen may be considered a concave curve 

with no apparent inflection point (fig. 2). Following their 

graphical examinations, the authors proposed that functions 

generating concave shapes and functions generating 

sigmoid shapes should be considered in the process of 

functional selection in order to get an accurate heightdiameter model. Thus, they selected and compared 20 

published, nonlinear height-diameter functions for 16 tree 

species in Alberta, Canada. Interestingly, based on the tstatistics for significance of parameters, mean squared error 

(MSE) values, and the plot of studentized residuals against 

the predicted heights, all the equations recommended for 

height-diameter relationships from their study produced Sshaped curves with inflection points. 

Fang and Bailey (1998) investigated 33 height-diameter 

equations, including S-shaped and concave-shaped curves, 

for tropical forests on Hainan Island in southern China. 

They assumed that height-diameter relationships could be 

described by height-growth curves with age replaced by 
I 

Asymptote 

Figure 1-Sigmoidal curve. 

I Asymptote 

Figure 2-Concave curve. 

d.b.h. They considered a number of characteristics in 

selecting functions but neglected functional inflection point 

as a selection criterion. Based on MSE values, most of the 

best equations with three parameters gave S-shaped curves 

with inflection points. 

Biological Growth Pattern Points to Desirable 

Mathematical Properties 

In addition to Huang and others (1992) and Fang and Bailey 

(1998), many others have used both S-shaped and concaveshaped curves in forest models for height-diameter curves, 

e.g., Arabatzis and Burkhart 1992. However it seems that in 

the course of model selection, none of these published 

papers considered the biological process of tree growth. 

From a biological point of view, a curve of height growth 

does exhibit a sigmoid or S-shaped pattern for a normal tree 

life span (Chaplain and others 1999, Clutter and others 

1983, Thompson and others 1992). From a mathematical 

perspective, curves for height growth should possess the 

properties of (1) monotonic increment, (2) inflection point, 

and (3) asymptotic value. These properties can be obtained 

by taking the first derivative of the dependent variable Cy) 

with respect to the independent variable (t) and setting it to 

be greater than zero, i.e., dy/dt > 0, for monotonic 

increment, by taking the second derivative to be equal to 

zero for inflection point, i.e., d2y/dt2 = 0, or by constraining 

the dependent variable to approach an asymptote as the 

independent variable goes to infinity, i.e., y-+ asymptote as 

t+m* 

Ambiguity 

Why were both the S-shaped and the concave-shaped 

functional forms always considered as competing models in 

the selection process? We think that some characteristics of 

such functions for height-diameter relationships and the 

biological process of tree growth are understood 

ambiguously, so that comparisons of functional forms often 

have been based on empirical criteria, i.e., data-related 

criteria, with little regard to the characteristics imposed a 

priori by the functional forms on the estimated curve. It is 

important to have evaluated and acquired an understanding 

of the assumptions and implicit limitations imposed by these 

commonly used models. This is because the inherent 

mathematical characteristics of a given functional form have 

important implications (1) for the hypotheses that can be 

tested, (2) for the statistical estimation methods employed, 

and (3) on the shape of the resulting height-diameter curve. 

Also, considerable bias can arise due to the use of an 

inappropriate functional form. 

Therefore, at least three important characteristics of heightdiameter curves should be examined before fitting a model: 

(1) monotonic increment, (2) inflection point, and (3) 

asymptote, e.g., Parresol 1992, p. 1,430. An S-shaped curve 

has the properties of monotonic increment, inflection point, 

and asymptotic value, while a concave-shaped curve has 

properties of monotonic increment and asymptotic value, 

but has no inflection point. Why was an S-shaped model 

better than a concave-shaped one in the reviewed papers? 

This is because those characteristics exhibited by the Sshaped models are more appropriate to describe a realistic 

height-diameter relationship that exhibits an S-shaped 

biological growth pattern. 

Previous height-diameter models may have been concave 

because only trees larger than the inflection point were in 
the database. In this case, the concave model may have 

better fit statistics than an S-shaped model, but it may be 

poor in the lower range where there are no data (poor 

extrapolation properties). Thus, the S-shaped model reflects 

appropriate bioIogica1 properties that are not captured by the 

data-driven concave model but may exhibit slightly poorer 

fit statistics. This may be why some researchers have felt 

that the concave model was superior. Today, however, when 

total biomass utilization and global carbon-uptake are 

predominant issues, we also are interested in smaller trees. 

Thus, the S-shaped models should be considered more 

appropriate than ever before. 

Recommendations 

In light of our reviews and discussions, we propose and 

recommend the following for modeling height-diameter 

relationships: 

1. The selection of a functional form for height-diameter 

relationships should not be restricted to the ease-of-fit to 

data, nor only to data-related criteria, but also should 

consider characteristics of the chosen model, such as 

monotonic increment, functional inflection point, and 

asymptotic value. These characteristics are very important 

(especially for making extrapolations) and easy to test for 

the chosen model forms-to see whether or not they are 

suitable for modeling height-diameter relationships. A 

functional form with monotonic increment, inflection 

point, and asymptotic value can describe height-diameter 

relationships appropriately. The curve of the functional 

form should be typical of a height cumulative growth 

curve, which starts at the origin value, increases steadily 

to attain maximum growth at an inflection point, and then 

gradually approaches an asymptotical value. This type of 

curve is also directly compatible with a height-increment 

curve, where the increment begins at a value of zero, 

increases steadily to reach a maximum, and then 

decreases asymptotically toward zero. 

2. Data for model building should include early height 

growth. Functions generating a concave-shaped curve as 

competing models to depict height over diameter should 

not be necessary. Generally, functions rendering a 

concave curve cannot describe tree or stand growth 

behaviors appropriately (especially fast-growth trees and 

stands). However, if a data set includes only larger or 

older trees beyond the inflection point, then a model 

generating a concave curve will probably work best. Even 

though a sigmoid model is correct theoretically, it may 

not give a good fit if the range of the data is beyond the 

point of inflection for the species and growing conditions. 

3. Sigmoid or S-shaped firnetions may be expanded to 

model other forestry relationships-such as volume-age, 

height-age, and diameter-age functions. 

4. The Bertalanffy-Richards growth function (Richards 

1959) and the Schnute (1 98 1) models are probably the 

most flexible and versatile functions available for 

modeling height-diameter relationships. Both are able to 

assume various shapes with different parameter values, 

and they produce satisfactory curves under wide-ranging 

biological/ecological modeling circumstances. The 

Schnute model is easy to fit and quick to achieve 

convergence for any database (Lei 1998). The 

Bertalanffy-Richards function is 

6, (1 - b2e-4DBH)i'/(l-b46,,1 9 

if b, > 1 or 0 5 b, < 1 or instances when b, c 0 

6, (b, ehDBH - l)["('-b4 11 3 

under certain conditions when 6, < 0 

The Schnute function is 

where 

DBHl = diameter of a young tree (lower range of data), 

DBH2 = diameter of an old tree (upper range of data), 

HI = tree height at DBH1, 

H2 = tree height at DBH2, 

a = constant acceleration in growth rate, and 

b =; incremental acceleration in growth rate. 

The coefficients to be estimated in the Schnute function are 

HI, H2, a, and 6, The values DBH1 and DBH2 are fixed 

and are normally taken to be the smallest and largest 

diameters in the data. 

Example 

Consider the 40 height-diameter pairs listed in table 1. 

These data are for Hbmaliunz racernosum Jaeq. growing in 

the Caribbean National Forest on the tropical island of 

Puerto Rico and represent a typical species growing in the 

subtropical wet life zone (Holdridge 1967). These data 

points are graphed in figure 3 along with the fitted 

sigmoidal Schnute function 
Table 1-Height-diameter pairs for 40 measured trees of 

the species Homliitm racemusum from the Caribbean 

National Forest on the island of Puerto Rico 

Tree D.b.h. Height Tree D.b.h. Height 

I3.b.h. = diameter at breast height. 

and the fitted concave Meyer function 

H = 1.3 + 23.7065(1- e0.0497 DBH 1 

R2 = 0.65, MSE = 2.28 m 

where 

RMSE = root mean squared error H~ - ki)2 /(n - p) , 

n -: number of observations, and 

p = number of function parameters. 

It is obvious from figure 3 that the sigmoidal curve is better 

suited to the data than the concave curve. The asymptote (A) 

and the inflection point (DBH*, H*) for the Schnute 

function are: 

DBH* = DBH, -t DBN, 

In sumary, data-related and reasonable biological criteria 

should be considered sirnultaneously for estimating forestry 

relationships in the process of selecting functions. 

0 10 20 30 40 50 60 70 80 90 

I3.b.h. (cm) 

Figure 3-Height-diameter scatter plot of 40 trees of Homaiium 

racernosum Jacq. from Puerto Rico showing a sigmoid curve (Schnute 

function) and a concave curve (Meyer hnction) through the data points. 
Acknowledgments 

We are grateful to Drs. Pacheeo Marques and Boris Zeide 

for their many helpful suggestions. Thanks also to Drs. 

Clark Baldwin and Stan Zarnocb for their reviews. 

Literature Cited 

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methods and model forms for estimating height-diameter relationships 

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Fang, Z.; Bailey, R.L. 1998. Height-diameter models for tropical forest 

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Parresol, B.R. 1992. Baldcypress height-diameter equations and their 

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