United States 

Department of 

Agriculture Method for Estimating Potential 

Forest Service Tree-Grade Distributions for 

Northeastern Forest 

Experiment Station Northeastern Forest Species 

Research Paper NE-670 

Daniel A. Yaussy 
Abstract 

Generalized logistic regression was useld to distribute trees into four potential tree 

grades for 20 northeastern species groqps. Potential tree grade is defined as the 

tree grade based solely on the length and amount of clear cuttings and defects, 

disregarding minimum grading diameters. The algorithms described use site 

index and tree diameter as the predictive variables, allowing the equations to be 

incorporated into individual-tree growth and yield simulators such as NE-TWIGS. 

The Author 

DANIEL A. YAUSSY, research forester with the Northeastern Forest Experiment 

Station at Delaware, Ohio, received a B.S. degree in natural resources (forestry) 

from the Ohio State University in 1976, an M.S. degree in forest biometrics from 

Virginia Polytechnic Institute and State University in 1978, and an M.S. degree in 

theoretical statistics from Ohio State in 1984. Since joining the USDA Forest 

Service in 1979, he has served on projects studying uneven-age management for 

small forest properties; tree, log, and lumber quality; growth, yield, and value 

development of northeastern forest types; and developing quantitative methods 

for modeling response of northeastern forest ecosystems to management and 

environmental stresses. 

Manuscript received for publication 25 September 1992 

Northeastern Forest Experiment Station 

5 Radnor Corporate Center 

100 Matsonford Road, Suite 200 

P.O. Box 6775 

Radnor, Pennsylvania 19087-4585 

March 1993 
, Introduction 

To quantify the value potential of a tree in a simulator, some 

form of quality must be assigned. Models have been 

developed that allow the distribution of butt-log grades over 

diameter at breast height (d.b.h.) (Ernst and Marquis 1979; 

Myers and others 1986; Dale and Brisbin 1985). This 

method provides a snapshot of quality and value, but does 

not indicate the potential of the trees. Lyon and Reed (1987) 

developed discriminant functions to assign tree grades to 

northern species, and provided additional discriminant 

functions which predicted future tree grades based on initial 

grades. These functions were incorporated into PROQUAL, 

an uneven-age stand-level simulator based on the SHAF 

model by Adams and Ek (1974). 

Potential grade (Gp) (Yaussy 1991a) is a variable that 

accounts for the change in tree quality over time. Yaussy 

(1991 b) describes a method to estimate the probability that 

a tree in an even-aged upland oak stand would be in one of 

four Gp classes. For this forest type, it was found that 

species, d.b.h., and stand age were the variables most 

correlated with Gp. But in expanding this research to 

different forest types with indeterminate age structures, it 

was apparent that another stand variable was needed. 

Using the same upland oak data set, correlations were 

calci~lated between Gp and basal area, site index, tree 

d.b.h. relative to quadratic mean d.b.h., and basal area in 

trees with larger diameters. Of the variables tested, site 

index had the highest correlation with Gp. Site index also is 

useful because it is unaffected by management practices. 

This paper describes the use of generalized logistic 

regression to estimate the probability distribution of Gp as a 

function of species group, site index, and d.b.h. Equations 

are presented for 20 species groups found in ageindeterminate stands of the Northeastern United States. 

Data Source 

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. Data were collected from 115-acre permanent plots in 

the most recent inventories of Kentucky, Maryland, 

Pennsylvania, and West Virginia. Table 1 lists the numbers 

of trees by species and the resulting 20 species groups 

used by NE-TWIGS. Site index (base age 50 years) was 

measured on each plot for the dominant species. For other 

species on the plot, conversion equations were used to 

assign the appropriate site index to each tree.' Table 2 lists 

the descriptive statistics for the variables of interest by 

species groups. Roughly 10 percent of the data set was 

randomly chosen for a validation data set. 

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

file at the Northeastern Forest Experiment Station, 

Delaware, OH. 
Table 1 .-Number of observations by species and species group 

Species Code Species No. in No. in 

WUP species group 

Ash ASH 

Basswood 

Beech 

Birch 

BAS 

BEE 

BIR 

Black cherry BLC 

Black oak BLO 

Chestnut oak CHO 

Commercial COH 

hardwoods 

Hemlock HEM 

Hickory HIC 

Noncommercial NOH 

hardwoods 

Black ash 

Green ash 

Pumpkin ash 

White ash 

American basswood 

White basswood 

American beech 

River birch 

Sweet birch 

Paper birch 

Yellow birch 

Black cherry 

Cherrylplum spp. 

Pin cherry 

Black oak 

Chestnut oak 

Swamp chestnut oak 

Chinkapin oak 

Post oak 

Buckeye spp. 

Catalpa 

Hackberry 

Persimmon 

Butternut 

Black walnut 

Sweetgum 

Magnolia spp. 

Cucumbertree 

Water tupelo 

Blackgum 

Paulownia 

American sycamore 

Eastern cottonwood 

Bigtooth aspen 

Quaking aspen 

Black willow 

Sassafras 

Elm spp. 

Atlantic white-cedar 

Baldcypress 

Balsam fir 

Eastern hemlock 

Eastern redcedar 

Red spruce 

Tamarack 

White spruce 

Hickory spp. 

Ailanthus 

American hdiy 

American hornbeam 

Apple 
Table 1 .-Continued 

Species 

group 

Code Species No. in No. in 

species group 

Noncommercial NOH 

hardwoods 

Northern red oak NRO 

Other red oaks OR0 

Other pines OTP 

Red maple REM 

Sugar maple SUM 

Virginia pine VIP 

White oak WHO 

White pine WHP 

Yellow-poplar YEP 

Balsam poplar 

Black locust 

Eastern hophornbeam 

Eastern redbud 

Honeylocust 

Kentucky coffeetree 

Mulberry spp. 

Osage-orange 

Serviceberry 

Sourwood 

Sugarberry 

Unknown spp. 

Northern red oak 

Blackjack oak 

Cherrybark oak 

Pin oak 

Scarlet oak 

Shingle oak 

Shumard oak 

Southern red oak 

Water oak 

Willow oak 

Austrian pine 

Loblolly pine 

Pitch pine 

Pond pine 

Red pine 

Scotch pine 

Shortleaf pine 

Table Mountain pine 

Boxelder 

Mountain maple 

Red maple 

Silver maple 

Striped maple 

Black maple 

Sugar maple 

Virginia pine 

Bur oak 

Swamp white oak 

White oak 

Eastern white pine 
Table 2.-Descriptive statistics for the development and validation data sets 

Species Development Validation 

group Variable N Mean Std. Dev. N Mean Std. Dev. 

ASH 

BAS 

BEE 

BIR 

BLC 

BLO 

CHO 

COH 

HEM 

HIC 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D,b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

0.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 
Table 2.-Continued 

Species Development Validation 

group Variable N Mean Std. Dev. N Mean Std. Dev. 

NOH 

NRO 

OR0 

OTP 

REM 

SUM 

VIP 

WHO 

WHP 

YEP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

G P 

D.b.h. 

Site index 

G P 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 

D.b.h. 

Site index 

GP 
Variable Definition 

Hardwoods 

USDA Forest Service hardwood tree grading standards 

include d.b.h. restrictions of 16 inches for a grade 1 tree 

and 13 inches for a grade 2 (Hanks 1976). Gp disregards 

these d.b.h. restrictions and surface defects that will 

disapear as the tree grows. Gp is defined as the actual 

Forest Service tree grade that a tree will attain when it 

grows into the 16-inch diameter class. When a tree enters 

the Ibinch class, Gp and actual tree grade will be identical. 

Gp is a discrete variable with four categories: grade 1, 

grade 2, grade 3, and below grade. Actual tree grade for 

hardwoods can be determined directly from Gp and d.b.h. 

For example, a tree with Gp of 1 and d.b.h. of 12.4 would 

have an actual grade of 3. As d.b.h. increases beyond the 

12.6- and 15.6-inch thresholds, the actual grade would 

change to 2 and then to 1. 

White Pine 

The white pine tree grades used by FIA require four full 

length clear faces for a tree to be considered grade 1 unless 

the d.b.h. is larger than 16 inches. This restriction would be 

relaxed for Gp determination. Actuai tree grade cannot be 

determined from white pine Gp since there can be grade 1 

trees less than 16 inches d.b.h. if they have four full-length 

clear faces. 

Other Pines 

All other pines were graded using the southern pine tree 

grading system which has no diameter restrictions. 

Therefore, Gp is identical to actual tree grade and the 

equations in this study partition the other pines into actual 

tree grades. 

Other Conifers 

Minimum merchantability standards were used to sort the 

spruce, fir, tamarack, and hemlock trees into two classes: 

merchantable and cull. The only diameter restriction 

imposed was that 3-inch knots were acceptable if the small 

end of the grading section was 13 inches or more (inside 

bark); otherwise, the knots must be 2 inches or less. 

Methods 

The proportions of trees in each Gp category based on 

d.b.h. and age were estimated by generalized logistic 

regression (GLR) as described by the CATMOD procedure 

of SAS (1985). With a discrete response variable and 

continuous predictor variables, the maximum likelihood 

procedure of logistic regression is indicated. Logistic 

regression is normally used with dichotomous response 

variables; however, GLR allows responses with more than 

two levels. 

Let pi denote the probability that the response equals i 

(i = 1, ..., r). A GLR model is of the form: 

where: 

fj = a function of predictor variables. 

If follows from (1) that: 

Note that the pi's must sum to 1, therefore: 

and r-1 

pr= 1/(1 + exp (fj)). 

j=1 

In the case of Gp being response variable and d.b.h. and 

site index being the predictor variables: 

pi = the probability that Gp equals i, i = 1,2,3, below grade. 

fj = bio + bjl *site index + bj,+d.b.h. + bj3*site index*d.b.h. 

bi, = regression coefficients to be determined. 

j =1,2,3. 

Results and Validation 

The coefficients and significance statistics that resulted 

from fitting the model are listed in Table 3. Site index and 

its interaction with d.b.h. were significant variables for 

species classified as intermediate or intolerant in shade 

tolerance. Moderately tolerant and very tolerant species 

usually are not dominant or codominant trees throughout 

their lives, which is a requirement for determining site 

index. Site index has little significance to these species; an 

exception is black cherry which is intermediate in tolerance, 

but the relationship between site index and Gp is 

insignificant. Many stands of cherry in the Allegheny 

Plateau of Pennsylvania have experienced extensive stem 

breakage associated with ice storms (Auchmoody and 

Rexrode 1984). This type of damage usually results in 

inaccurate measurements of site index. 
Table 3.-Coefficients and chi-square significance statistics for the generalized logistic 

regressions8 

Species Index Intercept Site D.b.h. D.b.h.* 

group j index Site index 

0 1 2 3 

ASH 

BAS 

BEE 

BLC 

BLO 

CHO 

COH 

HEM 

HIC 

NOH 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 
Table 3.--Continued 

Species Index Intercept Site D.b.h. D.b.h.* 

group j index Site index 

0 1 2 3 

NRO 

OR0 

OTP 

REM 

SUM 

VIP 

WHO 

WHP 

YEP 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

1 

2 

3 

p-value 

-- -- - 

aModel is of the form: 

pi = p, * exp (fj). 

where: 

f, = bjo + b,, *d.b.h. + b,, + d.b.h. + bj3 * siteindex * d.b.h. 

pi = the probability that Gp equals j, j = 1, 2, 3. 
Potential grade probabilities were calculated for each tree in 

the validation data set. A uniform random number was 

generated and a predicted Gp was assigned to each tree. 

Since these equations are to be used in simulators, I was 

not interested in how well they worked for individual trees, 

but on the resulting distribution of Gp's. Table 4 shows the 

actual and predicted totals for each species group and the 

level attained in a x2 test of goodness of fit (the larger the 

p-value, the better the fit). The equations performed 

adequately for most of the species groups in which 

stem quality is a concern. Surprisingly, the white oak 

and yellow poplar groups did not test as well as would 

have been expected from the significance of the coefficients 

shown in Table 3. Figures 1 a-b compare the Gp distributions 

for the development and validation data sets and the 

predicted Gp distribution for the validation set for white oak 

and yellow-poplar. Although the differences between the 

actual and predicted distributions of the validation set are 

not large or unreasonable, the large number of observations 

in the divisor of the X* statistic determines these differences 

to be important. 

0 Development 

Val idation 

Predicted Validation 

3 Below 

grade 

0 Development 

Validation 

a Predicted Validation 

3 Be l ow 

grade 

Figure 1 .- Percentages of trees for the development and validation data sets and for those 

predicted for the validation data set by the models for white oak (la) and yellow-poplar (1 b). 
Table 4.-Actual and predicted distributions of the validation data set and significance 

level attained by a x2 test of goodness of fita 

Species p-valuea GP Actual Predicted 

group 

COH 

HEM 

HIC 

NOH 

Bee 

BIR 

BLC 

BLO 

CHO 

ASH 0.183 

BAS 0.056 

0.737 

0.111 

0.348 

0.783 

0.778 

0.064 

0.005 

0.364 

0.425 

- 1 

2 

3 - Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

3 

3 

Below grade 

1 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 
Table 4.-Continued 

Species p-valuea GP Actual Predicted 

OTP 

REM 

SUM 

VIP 

WHO 

WHP 

YEP 

NRO 0.712 

OR0 0.009 

0.282 

0.483 

0.21 1 

0.265 

0.078 

0.691 

0.003 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

1 

2 

3 

Below grade 

aThe form of the statistic is: (actual-~redicted)~ 

x2 a I: predicted i= 1 

The larger the p-value, the more likely that the actual distribution of Gp is the 

same as the predicted distribution. 
Application Literature Cited 

The proportion of trees that will be in one of the four 

potential tree-grade classes is determined from equations 

(2) and (3). An example of the use of the equations for 

northern red oaks with a d.b.h. (D) of 14 inches and site 

index (S) 70 feet is: 

exp (f,) = exp (-2.772 + 0.0194 *S + 0.1 115 * 

D - 0.00085 * S * D) 

= .5036 

exp (f,) = 1.0045 

exp (fJ = 1.4518 

Z exp (fj) = 2.9599 

The proportion of trees of 'this species, d.b.h., and site 

index with a potential tree grade of below grade is (from 

equation (3)): 

From equation (2), the proportion of trees of this species, 

d.b.h., and site index with the other three potential treegrades is: 

In a growth and yield simulator such as NE-TWIGS, a 

uniform random number would be generated and potential 

tree grade would be assigned based on cumulative 

proportions. For example, if the random number fell 

between zero and 0.1272, the tree would be assigned a 

grade 1 ; between 0.1272 and 0.3809 (p, + p,), a grade 2; 

between 0.3809 and 0.7475 (p, + p, + pJ, a grade 3; and 

between 0.7475 and 1 below grade. 

Summary 

This study used FIA data on 20 species groups to develop a 

method to distribute trees into quality classes. This method 

relies on species, d.b.h., site index, and generalized logistic 

regression techniques to assign probabilities of being in one 

of four potential tree grade classes (two for nonpine conifers). 

The equations fit well for all but the shade-tolerant species 

for which site index has little meaning. Validation of the 

equations was performed using an independent data set 

also from FIA data. 

Acknowledgment 

I thank the Forest Inventory and Analysis unit of the 

Northeastern Forest Experiment Station for providing the 

data for this study. 

Adams, D.M.; Ek, A.R. 1974. Optimizing the management 

of uneven-aged forest stands. Canadian Journal of 

Forest Research. 4: 274-287. 

Auchmoody, L.R.; Rexrode, C.O. 1984. Black cherry site 

index curves for the Allegheny Plateau. Res. Pap. 

NE-549. Broomall, PA: U.S. Department of Agriculture, 

Forest Service, Northeastern Forest Experiment Station. 

5 P. 

Dale, M.E.; Brisbin, R.L. 1985. Butt log quality of trees in 

Pennsylvania oak stands. Res. Pap. NE-568. Broomall, 

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

Northeastern Forest Experiment Station. 8 p. 

Ernst, R.L.; Marquis, D.A. 1979. Tree grade distribution 

on Allegheny hardwoods. Res. Pap. NE-275. Broomall, 

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

Northeastern Forest Experiment Station. 5 p. 

Myers, J.R.; Miller, G.W.; Wiant, H.V. Jr.; Barnard, J.E. 

1986. Butt-log distributions for five Appalachian 

hardwood species. Res. Pap. NE-590. Broomall, PA: 

U.S. Department of Agriculture, Forest Service, 

Northeastern Forest Experiment Station. 4 p. 

Hanks, L.F. 1976. Hardwood tree grades for factory 

lumber. Res. Pap. NE-333. Broomall, PA: U.S. 

Department of Agriculture, Forest Service, Northeastern 

Forest Experiment Station. 81 p. 

Lyon, G.W.; Reed, D.D. 1987. A method for projecting 

stem quality and log grade distribution in sugar 

maple. In: Ek, A.R.; Shifley, S.R.; Burk, T.E., eds. Forest 

growth modeling and prediction; 1987 August 24-28; 

Minneapolis, MN. Gen. Tech. Rep. NC-120. St. Paul, 

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

North Central Forest Experiment Station: 1021 -1 028. 

SAS Institute, Inc. 1985. SAS user's guide: statistics, 

version 5 edition. Cary, NC: SAS Institute, Inc. 956 p. 

Yaussy, D.A. 1991a. Tree, log, and lumber quality 

models for Eastern hardwoods. In: Pacific rim 

forestry-bridging the world: Proceedings of the 1991 

Society of American Foresters national convention; 1991 

August 4-7; San Francisco, CA. Bethesda, MD: Society 

of American Foresters: 99-106. 

Yaussy, D.A. 1991b. Upland oak growth and yield 

simulator thinning rule influenced by grade. In: 

Payendeh, B., ed. Forestry futures: proceedings of 

Midwest mensurationists, Great Lakes forest growth and 

yield cooperative, and the Forestry Canada modeling 

working group joint workshop; 1991 August 20-23; Sault 

Ste. Marie, ON. Sault Ste. Marie, ON: Forestry Canada: 

61-68. 
Yaussy, Daniel A. 1993. Method for estimating potential tree-grade 

distributions for northeastern forest species. Res. Pap. NE-670. 

Radnor, PA: U.S. Department of Agriculture, Forest Service, Northeastern 

Forest Experiment Station. 12 p. 

Generalized logistic regression was used to distribute trees into four 

potential tree grades for 20 northeastern species groups. The potential tree 

grade is defined as the tree grade based on the length and amount of clear 

cuttings and defects only, disregarding minimum grading diameter. The 

algorithms described use site index and tree diameter as the predictive 

variables, allowing the equations to be incorporated into individual-tree 

growth and yield simulators such as NE-TWIGS. 

Keywords: Generalized logistic regression; NE-TWIGS; growth and yield 

simulators 

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Orono, Maine, in cooperation with the University of Maine 

Parsons, West Virginia 

Princeton, West Virginia 

Syracuse, New York, in cooperation with the State University of New York, 

College of Environmental Sciences and Forestry at Syracuse University 

University Park, Pennsylvania, in cooperation with The Pennsylvania State University 

Warren, Pennsylvania 

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