United States 

Department of I 

Agriculture 

i Forest Service 

i 

Nsfihsastern Forest ! Experiment Station i 

Research Paper NE-650 

el A. Yaussy 

in E. Dale 
MAWTlN E. DALE, 

Forest Service in 1 

1973, he received 

was assigned to the Norfheaster 

Laboratory at Delaware, Ohio. Currently, he is eel in research on the growth 

and yield of managed forest stands in the Northeast. 

Manuscript received for publication 2 Octo 

ortheastern Forest Experiment Station 

Wadnor, Pennsylvania 49087 

March 1991 

ide variation of merchantable sawlog length between two 

trees of same species, same size. 
Introduction 

Stem diameter and merchantable bole height are two key 

measurements needed for reliable estimates of tree volume, 

either in terms of cubic or board measure. Stand volume, of 

course, is an important stand attribute because it allows the 

manager to place an economic value on the timber, 

especially when one volume is known by species and size 

class. It is important to have accurate estimates of current 

volume or value of a stand, particularily at the time it is to 

be harvested or sold. However, even more important to the 

manager choosing from among several possible silvicultural 

options is the relative change in volume or value over time. 

If information can be developed that shows the relative 

change in volume or value as a result of applying each of 

several silvicultural treatments, then the manager has a 

very objective, rational basis for selecting the appropriate 

option. 

To help managers develop this type of information on stand 

volume changes, researchers have created various growthand-yield prediction models that simulate how trees and 

stands respond to different silvicultural treatments over 

time. Most of the programs that simulate individual-tree 

growth utilize regression equations that predict how much a 

tree of a given species grows in dimensions such as d.b.h. 

or height in relation to various tree and stand characteristics 

such as age, stocking, competitive status, site quality, and 

so on. This paper describes how the generalized 

merchantable-height equations used in the individual-tree 

growth-and-yield model NE-TWIGS (Hilt and Teck 1989) 

insufficient height data to develop merchantable-height 

equations for red pine and loblolly pine so they were 

combined with other pines as indicated in Table 2. 

Similarily, we combined paper birch and other birches with 

yellow birch because of insufficient height data. Our total 

sample consisted of more than 70,000 trees measured for 

bole length and about 30,000 trees measured for sawlog 

length (Table 2). 

Sawlog length and bole length as used here are defined by 

FIA in their data collection procedures as follows: 

Sawlog length is the measurement in feet from a 1-foot 

stump to the first of: 

a. minimum-top-sawlog diameter (9-inch-diameter outside 

bark (d.0.b.) for hardwoods and 7-inch d.0.b. for softwoods). 

Minimum length of sawlog is 8 feet. 

b. the point above which no log, whether or not 

merchantable, can be produced because of excessive 

limbs, forks, or crooks. 

c. on trees broken off, the point of the break. 

Bole length is the measurement in feet from a l-foot stump 

to the first of: 

a. a 4-inch-top d.0.b. Minimum length is 5 feet. 

b. the point where the central stem terminates by branching 

before reaching a 4-inch-top d.0.b. However, bole length 

can extend up through major subdivisions of the central 

stem. When this occurs, the measurement should follow the 

largest (in diameter) of the divisions. 

were developed. 

c. on trees broken off, the point of the break. 

Data for this study were collected by the Forest Inventory 

and Analysis (FIA) unit of the Northeastern Forest 

Experiment Station as part of the periodic survey of forested 

lands. They are a part of the data set used in developing the 

NE-TWIGS growth-and-yield model described by Teck 

(1990). Data were collected from more than 4,400 115-acre 

permanent plots during the 19601s, 70's, and 80's and 

included measurements from 14 Northeastern states. These 

survey plots covered a wide range of age, site, and stocking 

conditions. Basal area per acre ranged from 30 to 255 

ft21acre. Site index (base age 50 years) was measured on 

each plot for the dominant species and ranged from 30 to 

90 feet. For other species on the plot, we used conversion 

equations to assign the appropriate site index to each tree, 

depending on species. Quadratic-mean stand diameter 

ranged from 5 to 13 inches, which would indicate a wide 

range in the apparent age of size-class distribution of the 

sampled stands. 

Table 1 lists the 25 species groups used in this study. 

Although the NE-TWIGS program provides growth estimates 

for 28 separate species or species groups, there were 

'Teck, R. M.; Fuller, L. G.; Hilt, D. E. 1988. Untitled report, 

on file with authors. 

Variation in merchantable bole and sawlog height is great 

between individual trees even within a given diameter class. 

Figure 1 indicates the extent of this natural variation in 

sawlog length for chestnut oak. Among the factors 

contributing to this large variation in the merchantable 

height of trees within stands and between stands are 

species, site quality, and stand density. Other natural 

factors such as ice, wind, or other damage and humancaused disturbances such as type of harvest cutting can 

lead to great variation in merchantable tree height for a tree 

within a given species and d.b.h. class. A major source of 

variation in reporting merchantable heights is the difficulty 

in measuring them. The range in merchantable height is 

probably much larger than it would be in a specific stand 

because survey plots represent all types of stand 

conditions. Some plots represent severely high-graded 

stands where only the low-quality, forked, and cull trees 

remain, while some plots represent the very best stand 

conditions where the stand has been silviculturally treated 

to leave only the tallest, straightest, and highest quality 

stems. Table 2 shows the mean, maximum, and standard 

deviation for bole length and sawlog height of each species. 

The minimum-bole and sawlog lengths are not shown, but 

in most cases these were as defined, that is, an 8-foot 

minimum for sawlogs and a 5-foot minimum for bole length. 
Table 1 .-List of species, by species group, that were modeled 

Species group 

-- 

American beech 

Balsam fir 

Black cherry 

Black oak 

Chestnut oak 

Eastern hemlock 

Hickory 

Noncommercial 

hardwoods 

Northern red oak 

Northern white-cedar 

Other hardwoods 

American beech 

Balsam fir 

Black cherry 

Black oak 

Chestnut oak 

Swamp chestnut oak 

Eastern hemlock 

Hickory spp. 

Boxelder 

Striped maple 

Serviceberry 

Gray birch 

American hornbeam 

Eastern redbud 

Flowering dogwood 

Hawthorne 

Apple 

Ironwood 

Pin cherry 

Northern red oak 

Atlantic white-cedar 

Northern white-cedar 

Eastern redcedar 

American buckeye 

Yellow buckeye 

Hackberry 

Persimmon 

Honey locust 

American holly 

Butternut 

Black walnut 

Water tupelo 

Blackgum 

Sourwood 

Sycamore 

Black locust 

Black willow 

Sassafras 

Basswood 

American elm 

Slippery elm 

Other pines 

Species 

Quaking aspen 

Species group Species 

Red maple 

Red spruce 

Scarlet oak 

Sugar maple 

Tamarack 

Virginia pine 

White ash 

White oak 

White pine 

White spruce 

Yellow birch 

Yellow-poplar 

Loblolly pine 

Pitch pine 

Scotch pine 

Shortleaf pine 

Red pine 

Balsam poplar 

Bigtooth aspen 

Quaking aspen 

Cottonwood 

Red maple 

Black spruce 

Norwary spruce 

Red spruce 

Scarlet oak 

Southern red oak 

Swamp red oak 

Shingle oak 

Water oak 

Pin oak 

Willow oak 

Sugar maple 

Tamarack 

Virginia pine 

White ash 

Black ash 

Green ash 

Blue ash 

White oak 

Swamp white oak 

Bur oak 

Chinkapin oak 

Post oak 

White pine 

White spruce 

Birch species 

Yellow-poplar 

Sweetgum 

Cucumbertree 
Table 2.-Descriptive statistics of sample trees, by species - - - -- - -- .. - - 

Sawtimber trees '"Jl~wood! EE _ - ____ 

Species 

group 

Sample D.b.h. Sawlog length Sample D.b.h. Bole ieng;h 

size Mean maximum SD Mean maximum SD size Mean maximum SD Mean maximum SD 

American beech 

Balsam fir 

Black cherry 

Black oak 

Chestnut oak 

Eastern hemlock 

Hickory 

Loblolly pine 

Noncommercial 

N. red oak 

N. white-cedar 

Other hardwoods 

Other pines 

Paper birch 

Quaking aspen 

Red maple 

Red pine 

Red spruce 

scarlet oak 

i Sugar maple 

Tamarack 1 Virginia pine 

White ash 

White oak 

White pine 

White spruce 

Yellow birch 

Yellow-poplar 

-- 

aRed pine and lobiolly pine included in other pines due to insufficient data. 

bAll birches included in yellow birch due to insufficient data. 
D.B.H. (inches) 

Figure 1 .-Chestnut oak sawlog lengths for all sites. The solid line represents the fitted 

equation with site index set at 60. 

Methods 

The Richards growth model (Richards 1959) frequently has 

been modified and used to fit total height of site-index trees 

as a function of site and stand age (Lundgren and Dolid 

1970; Beck 1971). Ek (1971) obtained an excellent fit for 

white spruce using the expression: 

where: 

H = total height of dominant and codominant trees in feet 

S = site index (height at age 50) 

A = age in years 

b's = constant parameters of the model. 

This model, referred to as Model 1, also has ~iven excellent 

results when used to fit data read directly from site-index 

curves for many other species (Payandeh 1974: Hilt and 

Dale 1982). 

One of the problems we encountered when using Model 1 

was the use of stand age as a predictor variable because 

many of the timber stands in the Northeastern United States 

contain trees of several age classes. Because of unevenaged stand conditions or the wide variation in tree ages 

within a stand, it is almost impossible to determine stand 

age with any degree of precision; hence, stand age was not 

used as a predictor variable in the NE-TWIGS growth-andyield modeling effort. So, for the merchantable-height 

models, tree diameter was substituted for age. Upon 

plotting our merchantable-height data, it did not appear 
to be doubly asymptotic in form as implied by Model 1, For each species group we used the merchantable-bole 

therefore, we tried the simplified form length or sawlog data to fit Model 2, using the nonlinear 

procedures provided by the Statistical Analysis System 

H = b I Sb2(1 - EXP(b3(D - TD))) (2) (SAS) (SAS Institute 1982). 

where: 

Results 

H = the merchantable height, either sawlog or bole length 

D = diameter at breast height (d.b.h.) outside bark at a The nonlinear regression coefficients computed for Model 2 

point 4.5 feet above the ground, in inches are given for each species in Table 3 for bole length and 

TD = the minimum top diameter outside bark, as stated in sawlog length. The species groups are listed as they appear 

the definitions above; that is, 9 inches for hardwood in the NE-TWIGS growth-and-yield simulator, so where the 

sawlogs, 7 inches for softwood sawlogs, and 4 inches data were pooled for some species, the same set of 

for pulpwood for all species. coefficients is listed more than once. 

Table 3.-Coefficients for predicting sawlog and bole length for northeastern species 

Species Sawlog length Indexa Bole length 

group b1 b2 b3 of fit b I b2 b 3 

American beech 

Balsam fir 

Black cherry 

Black oak 

Chestnut oak 

Eastern hemlock 

Hickory 

Loblolly pine 

Noncommercial 

N. red oak 

N. white-cedar 

Other hardwoods 

Other pines 

Paper birch 

Quaking aspen 

Red maple 

Red pine 

Red spruce 

Scarlet oak 

Sugar maple 

Tamarack 

Virginia pine 

White ash 

White oak 

White pine 

White spruce 

Yellow birch 

Yellow-poplar 

(corrected total sums of squares - residual sums of squares) alndex of fit = (corrected total sums of squares) 

b~odel form: Length, = bl Slb2(1 - EXP(b3(DBH - TD))), where i = sawlog, or bole. 

SI = site index, DBH = outside bark diameter at 4.5 feet, TD = the appropriate top diameter. 

Index 

of fit 

0.468 

0.621 

0.388 

0.439 

0.345 

0.608 

0.613 C 

Clndicates species that were combined due to insufficient data. The combinations are: 

1. white spruce, red spruce 

2. red pine, loblolly pine, other pines 

3. yellow birch, paper birch. 
The indices of fit, commonly known as the calculated R2, 

are appreciably higher for the bole-length equations than 

are those for sawlogs. This is partially because bole length 

is more closely related to total height, which depends on 

site and age or size class. Also, a better fit was expected 

because, by definition, the merchantable bole length can 

extend up through major forks or subdivisions of the central 

stem, whereas major branches, forks, or crooks limit sawlog 

length. In general, our model performed somewhat better 

for the coniferous species than for most hardwoods for both 

sawlog and bole-length predictions. This was not unexpected 

since conifers generally are more excurrent in form than 

hardwoods. Tulip-poplar and hickory are exceptions since 

they usually have an undivided stem, and this characteristic 

is evident in the higher index of fit for these two hardwoods. 

This compares favorably with most conifers. 

Although these merchantable-height equations fail to 

explain a large portion of the variation, they do provide 

useful predictions for many purposes. They are consistent 

with well-known trends that show merchantabe height 

increasing as d.b.h. and site index increase (Fig. 2). When 

used in growth models such as NE-TWIGS, along with 

diameter growth and volume equations, the manager can 

obtain estimates of the relative change in volume or value 

for a wide range of possible silvicultural treatments. This 

relative change in volume from the present to some future 

date, with and without the proposed silvicultural treatment, 

provides substantive support for the appropriate decision. 

Users are cautioned, however, that our merchantable-height 

equations probably should not be used to estimate present 

volume of a specific stand, especially if the volume estimate 

is the basis of a sale or a logging contract. In such cases we 

D.B.H. (inches) 

Figure 2.-Form of the bole-length equation fitted with specified site indices for chestnut oak. 
would recommend traditional inventory procedures that 

entail measuring every tree or a sample of trees for stem 

diameter and estimating or measuring stem height to either 

a fork or some minimum top diameter that restricts 

merchantability. Timber cruise software is available that 

uses these measurements to determine present volumes 

and values (Yaussy and Brisbin 1990). Remember, the 

merchantable-height equations presented here are regional 

averages and, as such, they may not apply to a specific 

stand since individual timber stands may vary widely from 

location to location, as noted earlier. 

Literature Cited 

Beck, D. E. 1971. Height-growth patterns and site 

index of white pine in the southern Appalachians. 

Forest Science. 17: 252-260. 

Ek, A. R. 1971. A formula for white spruce site index 

curves. University of Wisconsin For. Res. Note 161. 

Madison, WI: University of Wisconsin. 2 p. 

Hilt, 0. E.; Dale, M. E. 1982. Height prediction 

equations for even-aged upland oak stands. Res. Pap. 

NE-493. Broomall, PA: U.S. Department of Agriculture, 

Forest Service, Northeastern Forest Experiment Station. 

9 P. 

Hilt, D. E.; Teck, R. M. 1989. NE-TWIGS: An individualtree growth and yield projection system for the 

Northeastern United States. The Compiler. 7(2): 10-16. 

Lundgren, A. L.; Dolid, W. A. 1970. Biological growth 

functions describe published site index curves for 

Lake States timber species. Res. Pap. NC-36. St. Paul, 

MN: U.S. Department of Agriculture, Forest Service. 

North Central Forest Experiment Station. 9 p. 

Payandeh, B. 1974. Formulated site index curves for major 

timber species in Ontario. Forest Science. 20: 143-144. 

Richards, F. J. 1959. A flexible growth function for 

empirical use. Journal of Experimental Botany. lO(29): 

290-300. 

SAS Institute, Inc. 1982. SAS user's guide: statistics, 

1982 edition. Carey, NC: SAS Institute, Inc. 584 p. 

Teck, R. M. 1990. NE-TWIGS 3.0: An individual-tree 

growth and yield projection system for the 

Northeastern United States. The Compiler. 8(1): 25-27. 

Yaussy, D. A,; Brisbin, R. L. 1990. User's guide to STUMP: 

a system of timber utilization and mill processing. 

Gen. Tech. Rep. NE-138. Radnor, PA: U.S. Department 

of Agriculture, Forest Service, Northeastern Forest 

Experiment Station. 24 p. 
Yaussy, Daniel A.; Dale, Martin E. 1991. Merchantable sawlog and bolelength equations for the Northeastern United States. Res. Pap. 

NE-650. Radnor, PA: U.S. Department of Agriculture, Forest Service, 

Northeastern Forest Experiment Station. 7 p. 

A modified Richards growth model is used to develop species-specific 

coefficients for equations estimating the merchantable sawlog and bole 

lengths of trees from 25 species groups common to the Northeastern United 

States. These regression coefficients have been incorporated into the 

growth-and-yield simulation software, NE-TWIGS. 

ODC 561.2:242(74) 

Keywords: Nonlinear regression; individual tree models 

2 U. S. GOVERNMENT PRINTING OFFICE: 1991 /548-O88/2OOl5 
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