United States

Department of

Agriculture

Forest Service

A Tree Taper Model Based on Similar

Triangles and Use of Crown Ratio as a

Measure of Form in Taper Equations

for Longleaf Pine

Southern

Research Station

General Technical

Report SRS-66

Dennis J. Shaw, Ralph S. Meldahl, John S. Kush,

and Greg L. Somers
The Authors

Dennis J. Shaw, Graduate Research Assistant, Auburn University,

School of Forestry &Wildlife Sciences, Auburn University, AL 36849,

now resides in Lanett, AL 36863; Ralph S. Meldahi, Associate Professor,

John S. Kush, Research Associate, Greg L. Somers, Associate Professor,

Auburn University, School of Forestry & Wildlife Sciences, Auburn

University, AL 36849.

June 2003

Southern Research Station

p.0. Box 2680

Asheville, NC 28802
A Tree Taper Model Based on Similar Triangles

and Use of Crown Ratio as a Measure of Form in

Taper Equations for Longleaf Pine

Dennis J. Shaw, Ralph S. Meldahl,

John S. Kush,’ and Greg L. Somers

Abstract

WC used data from 322 natural longlcaf pine (Pinus pdustris Mill.)

trees to include crown ratio as a continuous variable in taper

equations. The data were divided into 10 crown-ratio classes and

fitted taper equations into each class to detect trends in the

coefficients. For application to longleaf pine, we replaced

coefficients that exhibited a trend with crown ratio with a function

of crown ratio. The inclusion of crown ratio as a continuous variable

improved by at least 16 percent the mean square residual for both

models. The authors’ model performed better on the modeling

dataset based on fit statistics and on the validation dataset. It also

contained fewer parameters and was easier to rearrange to solve for

height to a given diameter.

Keywords: Crown ratio, longleaf pine, Pinus pulusfris Mill.,

tree taper equations, tree taper model.

Introduction

Taper has been defined as the change in stem diameter

between two measurement points divided by the length of

the stem between these two points (Morris and Forslund

1992). Taper equations attempt to describe it as a function of

tree variables such as diameter at breast height (d.b.h.), total

height, etc. Many forest scientists and foresters have

demonstrated the importance of taper functions. According

to Kozak (1988), taper equations can be used to provide: (1)

predictions of inside bark diameters at any point on the

stem; (2) estimates of total stem volume; (3) estimates of

merchantable volume and merchantable height to any top

diameter and from any stump height; and (4) estimates of

individual log volumes.

Bruce and others (1968) believed that the inclusion of some

measure of tree form, e.g., crown ratio (CR), as an additional

independent variable might improve the fit of taper

equations. Dell (1979), Feduccia and others (I 979), and

Baldwin and Polmer (198 1) also discussed the potential of

using CR to improve stem-taper curves. CR is defined as the

percent of a tree’s total height occupied by the live crown.

’ Corresponding author

CR can be calculated as the ratio of live-crown length to total

tree height.

Several studies have explored the possibility of including CR

in taper equations, although not all have had positive

results. Valenti and Cao (1986) modified the three-segment

taper equation introduced by Max and Burkhart (1976) to

describe stem profile:

d2 = D”[p,z+p,z” +/qz-a,y1, +j?4(Z-a2)2/2]+E (1)

where cl = diameter at height h, = d.b.h., z = 1 - (WH), H =

total height of the tree, ai represents join points estimated

from the data (a, = upper join point, a2 = lower), the pi ‘s

are the model parameters estimated from the data, Ii = 1 if

z > ai (i= 1,2), or 0 otherwise, and E represents stochastic

error. This equation will be referred to as model VC. The

authors used data from a loblolly pine plantation and

divided the data into 10 CR classes with approximately

equal numbers of observations per class. Equation 1 was

fitted to each CR class individually, and the resulting

coefficients were plotted against the CR class medians to

detect trends. Based on these plots, some of the coefficients

were expressed as functions of CR. The resulting equation

had the form:

cl2 = D2[(yo + y, I CR) - z + /3,z”

where I, = 1 if z > y2 + y,ln (CR), or 0 otherwise, and I, = 1 if

z > a,, or 0 otherwise. The results of Valenti and Cao ( 1986)

indicated that the addition of CR to the model was a

significant improvement.

Our first objective was to determine if inclusion of CR as a

continuous variable significantly could improve taper

estimates for natural longleaf pine (Pinuspalustris Mill.). As

we examined various models, we noted problems with

residuals, especially for the portion of the bole below breast

height. To possibly remedy this problem, a new model was

developed to better address that portion of the bole below

breast height. This resultant model, along with Valenti and

Cao’s (1986) model, were then studied for inclusion of CR.
Data

Data for the modeling component of this study comprise two

sets. The larger set, used in a taper study by Farrar (1987),

consisted of 2,832 diameter measurements from 2 14 felled

longleaf pine trees from naturally regenerated, even-aged

stands in northwest Florida, southwest Georgia, central and

south Alabama, and south Mississippi. Single-stemmed trees

were selected over a range of diameters, total heights, and

CR classes; all were free of excessive crook or sweep and

visibly undamaged (Farrar 1987). Measurements included

diameter outside bark (d.o.b.), bark thickness, and height to

the diameter. Diameters were measured at I -inch taper steps

from a 0.2-foot stump.

The second datasct consisted of 1,376 diameter

measurements on 108 felled longleaf pine trees from evenaged, naturally regenerated stands in Alabama and Florida

that comprise a subset of plots in the USDA Forest Service

Regional Longleaf Pine Growth Study (Kush and others

1987). Trees from all crown classes were selected following

criteria of the Farrar study. Field measurements for each tree

included crown class, d.b.h. to nearest 0.1 inch and total

height and height to the base of the live crown to the nearest

0.1 foot. D.o.b. (using diameter tape), bark thickness, and

height to the point where the diameter was measured were

taken at the stump, between stump and breast height, at

breast height, and above breast height at 5-foot (+/-)

intervals to the tip of the tree. Length of the interval was

adjusted as necessary to avoid limbs, knots, or other defects

that abnormally affected the diameter.

Two independent datasets were obtained to conduct model

validation. The first dataset contains measurements on 59

felled longleaf pine trees from naturally regenerated stands

across southern Alabama. This dataset contains 684

diameter measurements.’ The second model-validation

dataset contains measurements of 33 felled longleaf pine

trees from a site in southwest Alabama. It contains 49 1

diameter measurements.2

Model Development

We developed a new taper mode1 by starting with a linear

mode1 containing three submodels represented by three

‘Glover, Glenn, 1979. Unpublished dataset. On tile with Auburn

University, School of Forestry & Wildlife Sciences, Auburn

University, AL 36849.

‘Meldahl, Ralph S., 1982. Unpublished dataset. On file with Auburn

University, School of Forestry & Wildlife Sciences, Auburn

University, AL 36849.

triangles (fig. I). The tree outline is represented by the

shaded polygon ILMBNOK. Point B would occur at the

tree’s total height, H. Point F would be at a height of H plus

a constant, y, . Point J would be at a height of H minus a

constant, y2. The upper join point, a,, would be

represented by a line through points M and N. The lower

join point, a,, would be represented by a line through

points L and 0.

Upper stem diameters were related to d.b.h. by using the

properties of similar triangles. Join points, a, and a,, are in

terms of relative height (/r/H). For a diameter, d, occurring

above the upper join point, a, (using triangle ABC), the

relationship to d.b.h. is given by:

dl(P,D)=(H-h)l(H-4.5) (3)

where h is the height (foot) to diameter d (inch), and pi is a

constant to account for the difference in the widths of

triangles ABC and EFG at breast height.

For a diameter occurring between a, and a, (using triangle

EFG), the relationship is given by:

dlD=(H+y,-h)l(H+y,-4.5) (4)

For a diameter occurring below the lower join point, a, ,

(using triangle IJK) the relationship is given by:

d/(/3, D) = (H - y,-h)l(H-YZ-4.5) (5)

d.b.h

AI E G KC

Figure i-Linear model of a tree

2
where j3, is a constant to account for the differences in widths of triangles IJK and EFG at breast height.

Intercepts, to allow models to be continuous at the join points, and stochastic errors were added, coefficients renumbered, and H

factored from the numerators to yield the model:

p, + /3’(H(l -x) l(H - 4.5)) + & lkxka,

/3,+/3,(H(l+y,/H-x)/(H+y,-4.5))+~ a, 2x>a,

/3~+j3p,(H(1-y,/H-x)l(H-y,-4.5))+~ a,>x>O

(6)

wherex is the relative height (WH) of the diameter, cl.

As we expected, the linear mode1 was not satisfactory for describing tree taper, so exponents were added to the second term in

each of the three segments. The exponents increased flexibility in the model as opposed to forcing a linear relationship on the

model. This resulted in:

/3,+P,(H(l-x)/(&4.5))”+E

~,+~3(H(1+y,/H-x)l(H+y,-4.5))q2+~

P,+P,(H(l-y,/H-x)l(H-~~-4.5))‘~ +E

12xZa,

a, >x2a,

a,>x20

(7)

Imposing the condition that the diameter at the tip of the tree equals zero (d = 0 when x = 1) results in /?,, equaling zero. Further,

imposing the conditions that the mode1 be continuous at the join points, and that the models have continuous first partial

derivatives with respect to x at the join points, allowed /3,, /3,, /?,, and /?, to be solved for in terms of fl,.

The resulting mode1 had the form:

dlD=/3,

i

~2(H(1+~-a,))q2~‘(H(l-x)))”

q,(H +y, -4.5)‘* (H (l-a,))‘lm’

+&

for 1 > x > a,,

d,D= v2P3 (N(l-a,))(H(l+~-a,))‘“‘-”

q(H+y, -4.572

for a, >.x>a,, and

63)

(9)

3
for a* >x>o.

rilD=/3,

~*(H(l-a,))(H(l+~-a,))‘~2-”

q(H +y, -4.5)“2

-4 i

rl,(H(l - $-a,)(H(l + +x~)~“~-‘)

Q(H +y, -4.5)‘2

j

In fitting this model to the data, a value for y2 was obtained that was not significantly different from zero (95-percent confidence

interval). y2 was replaced with zero, and the resulting model will be referred to as model A.

Methodology

The results from fitting mode1 VC (equation 1) and model A to the modeling data are given in table 1. The criteria used to

select the best mode1 form were: (1) bias (l/n) C(y, -pi), (2) relative bias (l/n) C(y, -Yi)ly, (3) absolute bias (lln)Clyj -?,I,

(4)absoluterelativebias(l/n) C/y,-?,I/y,,and(5)fitindex ~-~(~,-~,)~/~(~~i;)*;index 1-C(y,-~~)2/C(yi-;G,)2;where

yi and pi are the observed and predicted diameters for the ilh observation and y IS the average of yi values, respectively, and

n is the number of observations.

Table l-Fit statistics for models A, A/CR, VC, and VC/CR for inside bark

and outside bark diameters

Absolute

Absolute Relative relative Fit

Model” Parameters Bias bias bias bias index

A

A/CR

kc

VUCR

A

A/CR

kc

VCICR

Number

0.01651

-0.01272

0.00797

-0.00274

-0.00240 0.45670 0.002 13 0.08653 0.98464

0.00567 0.41938 O.Ol150 0.08339 0.98698

0.02166 0.46723 0.009 17 0.08604 0.98366

0.02012 0.42620 0.00940 0.07769 0.98615

Inside bark

0.4 1348 0.00922

0.36556 0.00324

0.41912 0.00304

0.37229 0.00292

Outside bark

0.09343 0.98369

0.08105 0.98680

0.09423 0.98336

0.08053 0.98625

0 Model A = the authors’ model (see equations 8, 9, and IO); model A/CR = the authors’

model incorporating crown ratio (see equations 12, 13, and 14); model VC = Valenti and

Cao’s model (see equation 1); model VCKR = Valenti and Cao’s model incorporating

crown ration (see equation 11).
The models then were examined for inclusion of CR as a

continuous variable following the approach used by

Valenti and Cao (1986). The modeling data were divided

into 10 CR classes having approximately the same number

of trees. The models were fitted separately to each CR

class. The resulting coefficients then were plotted against

CR class to see if any trends were observable. The plots

for model VC indicated that the most obvious relationships

with CR existed for /?,, /?,, and a,. The functions of

CR found to give the best fit with the data were

p, = y, + y&R, p, = y3 + y4CR2 , and a, = p, + p,CR . The

resulting model (VCKR) had the form:

d* =D*(fl,~+(y,+y~CR)z*

+(Y, +~;,cR*)(z-(P, +p?CR))*4

+P4(z-n2)212 (11)

The plots for model A indicated that the most significant

relationships with CR existed for parameters Q and a, . The

functions of CR that were found to give the best fit with the

data were q72 = 4 i- i$CR and a, = p, + p2CR2. The

resulting model (A/CR) had the form:

dlD=&

AB”-‘(H(l-x))

F”

dlD=&

(A-1)B” +(H(l-X))A

FA 1

for (p, + p2CR2) > x > a,, and

(14)

for oz2 >.x>O, whereE=H(l-a,).

Validation of models A/CR and VCKR was conducted using

the independent dataset described above. The criteria used

for comparing the models included model bias, or average

residual [C (predicted -0 h served) / n] , accuracy, or average

absolute residual (C [predicted - obsewedl/ n) , and rootmean-squared prediction error, or mean square residual

(MSR) (square root of C [ (predicted - ob.sewed)* / 111,

where n is the number of observations. Bias, which measures

expected error when several observations are to be

combined by averaging or totaling, and accuracy, which

measures the average error associated with the prediction of

any one observation, provides nearly all the information

necessary in validation (Burk 1986). Root-mean-squared

prediction error, which is equal to the standard deviation of

prediction error for unbiased models, is also recommended

as validation criteria (Burk 1986). The results of models A,

VC, A/CR, and VCKR on the validation dataset are found in

table 2. Parameter estimates for model A/CR for d.i.b. and

d.o.b. are found in table 3.

Conclusions

The addition of CR as a continuous variable in both models

A and VC was found to improve estimates of upper stem

diameters. Model A/CR had a decrease in MSR of 20.5

percent when CR was included. Model A/CR also had a

higher Rz2 value, a lower average residual, and a lower

average absolute residual, compared to model A [R,* is the

adjusted R2 (Draper and Smith 198 I)]. The idea is that the Ra2

can be used to compare equations fitted not only to a

specific set of data but also to two or more entirely different

sets of data. Inclusion of CR in model VC resulted in an

18.25-percent decrease in MSR, an increase in Ra2, a decrease

in average residual, and a decrease in average absolute

residual.

Although no significant differences in residual patterns were

noted, model A was found to outperform model VC in terms

of all four criteria on the modeling data for both d.i.b. and

d.o.b. (table 1). Both models showed significant

improvement with the addition of CR. Model A/CR was

found to outperform model VUCR in all four criteria for both

d.i.b. and d.o.b. (table 1). Tests on the validation dataset

resulted in both models again showing significant

improvement with the addition of CR. Model A/CR was

found to perform better than model VCYCR for both d.i.b. and

d.o.b.

Further, while model A/CR has a complex final form, it

contains fewer parameters than model VC/CR (7 vs. 9) and is

derived from a simpler concept. In addition, this equation is

easier to rearrange to solve for height to a given diameter

because only linear functions of height (h) are used (see

appendix). Rearranging model VCKR to solve for h results

in a quadratic equation.

5
Table 2-Validation statistics for models A, A/CR, VC, and VCICR for inside

bark and outside bark diameters

Absolute

Absolute Relative relative Fit

Model” Parameters Bias bias bias bias index

Number

Inside bark

A 7

A/CR 7

w 6

VUCR 9

-0.50935 0.63858 -0.08 128 0.11084 0.95703

-0.36443 0.50575 -0.05642 0.092 17 0.97302

-0.51034 0.64742 -0.07836 0.1 1061 0.95554

-0.39525 0.53092 -0.06133 0.09352 0.97027

Outside bark

A 7 -0.24684 0.47683 -0.04 10 1 0.083 19 0.97 152

A/CR 7 -0.10336 0.36799 -0.02788 0.07110 0.98233

kc 6 -0.19640 0.46690 -0.04 144 0.08442 0.97238

VCICR 9 -0.14503 0.39994 -0.03 156 0.07382 0.97975

’ Model A = the authors’ model (see equations 8, 9, and 10); model A/CR = the authors’

model incorporating crown ration (see equations 12, 13, and 14); model VC = Valenti

and Cao’s model (see equation 1); model VCKR = Valenti and Cao’s model incorporating

crown ration (see equation 11).

Table 3-Parameter estimates, asymptotic standard errors, and

confidence intervals for model A/CR for inside bark and outside bark

diameters

95-percent

Asymptotic confidence interval

Parameter Estimate standard error Lower Upper

Inside bark

5 0.07 194

p, 2.17988

q.i 48.92647

PI 0.89646

6 -0.38504

% 0.13536

hZ 0.12553

0.00208 0.06787

0.27950 1.63 I90

1.83986 45.31928

0.00600 0.88470

0.01 157 -0.40773

0.01372 0.10845

0.01328 0.09949

Outside bark

0.07602

2.72786

52.53366

0.90822

-0.36235

0.16227

0.15 157

3 0.07544 0.00181 0.07 190 0.07898

4 1.42627 0.04070 1.34648 1.50607

T 46.90098 1.48181 43.99578 49.80618

PI 0.93740 0.00443 0.92872 0.94608

6 -0.28840 0.01270 -0.3 1330 -0.2635 1

0.2568 1 0.01 169 0.23389 0.27973

0.23635 0.01255 0.21175 0.26095

Model A/CR = the authors’ model incorporating CR (see equations 12, 13, 14);

@,, n,, c(~, h,, hz, p,, and pz = regression coefficients.
Acknowledgments

The authors thank Bruce Zutter and Mark Dubois for their

helpful comments. Special acknowledgment is given to

Robert M. Farrar, Jr., for his efforts to design and establish

the continuing long-term Regional Longleaf Pine Growth and

Yield Study. Thanks are extended to Glenn Glover for the use

of his dataset,AIabamaAgricultural Experiment Station

Series No. 9-985957.

Research support was provided through the project

“Regional Longleaf Growth Study-30Year Remeasurement”

of Cooperative Agreement 19-92- 13 1 dated 1992-97. This

research was supported by funds provided by the U.S.

Department ofAgriculture, Forest Service; Southern

Research Station, Asheville, NC.

Literature Cited

Baldwin, V.C., Jr.; Polmer, B.H. 1981. Taper functions for unthinned

longleaf pine plantations on cutover West Gulf sites. In: Barnett,

James P., ed. Proceedings of the first biennial southern silvicultural

research conference; 1980 November 6-7; Atlanta. Gen. Tech.

Rep. SO-34. New Orleans: U.S. Department of Agriculture, Forest

Service, Southern Forest Experiment Station: 156-163.

Bruce, D.; Curtis, R.O.; Vancoevering, C. 1968. Development of a

system of taper and volume tables for red alder. Forest Science. 14:

339-350.

Burk, T.E. 1986. Growth and yield model validation: Have you ever

met one you liked‘? In: Allen, S.; Cooney, T.M., eds. Data

Management Issues in Forestry: Proceedings of a computer

conference and third annual meeting of the Forest Resources

Systems Institute; Florence, AL: Forest Resources Systems

Institute: 35-39.

Dell, T.R. 1979. Potential of using crown ratio in predicting product

yield. In: Frayer, WE., ed. Forest Resource Inventories: Workshop

Proceedings, Fort Collins, CO: Colorado State University,

Department of Forestry and Wood Science: 843-851, Vol. 2.

Draper, N.R.; Smith, H. 1981. Applied regression analysis. 2” cd. New

York: John Wiley. 709 p.

Farrar, R.M., Jr. 1987. Stem-profile functions for predicting multipleproduct volumes in natural longleaf pines. Southern Journal of

Applied Forestry. ll(3): 161-167.

Feduccia, D.P.; Dell, T.R.; Mann, W.F. [and others]. 1979. Yields of

unthinned loblolly pine plantations on cutover sites in the West

Gulf region. Res. Pap. SO-148. New Orleans: US. Department of

Agriculture, Forest Service, Southern Forest Experiment Station.

88 p.

Kozak, A. 1988. A variable-exponent taper equation

Journal of Forest Research. 18: 1363-1368.

Canadian

Kush, J.S.; Meldahl, R.S.; Dwyer, S.P.; Farrar, R.M., Jr. 1987.

Naturally regenerated longleaf pine growth and yield research. In:

Phillips, D.R., camp. Proceedings of the fourth biennial southern

silvicultural research conference; 1986 November 4-6; Atlanta.

Gen. Tech. Rep. SE-42. Asheville, NC: U.S. Department of

Agriculture, Forest Service, Southeastern Forest Experiment

Station: 343-344.

Max, T.A.; Burkhart, H.E. 1976. Segmented polynomial regression

applied to taper equations. Forest Science. 22: 283-288.

Morris, D.M.; Forslund, R.R. 1992. The relative importance of

competition, microsite and climate in controlling the stem taper

and profile shape in jack pine. Canadian Journal of Forest

Research. 22: 1999-2003.

Valenti, M.A.; Cao, Q.V. 1986. Use of crown ratio to improve loblolly

pint taper equations. Canadian Journal of Forest Research. 16:

1141-l 145.

7
Appendix

Equation A/CR rearranged to solve for height, h, to a given diameter, d

I

h =H

i

(PP’“P A

D

-(A- NBA)

i

I

((qJd(F”) EC”‘-‘) (vl) E(“-A)

)[(A-I) BA+H(I-a2jA/+ Eq’

D.7

(PjPA - A 1

l’or(&)D(E”+(A-l)B”)/F” <d<(,l&)DA +(p)~ (A-I)B”+E”]IF”-(P,)DA(E~~(~~,F~)). 3 [

where

B = H(l-p,-p,CR),

E = H(l-cq),

F = H-4.5,

D = d.b.h.,

H = total height of tree,

CR = crown ratio,

d = diameter at a given height, h, and

P3, 573, a2, 4, 47 PI9 and p2 are regression coefficients.
Shaw, Dennis .J.; Meldahl, Ralph S.; Kush, John S.; Somers, Greg L. 2003. A Tree

taper model based on similar triangles and use of crown ratio as a measure of form in

taper equations for longleaf pine. Gen. Tech. Rep. SRS-66. Asheville, NC: U.S.

Department ofAgriculture, Forest Service, Southern Research Station. 8 p.

We used data from 322 natural longleaf pine (Pinuspallrstris Mill.) trees to include crown

ratio as a continuous variable in taper equations. The data were divided into 10 crownratio classes and fitted taper equations into each class to detect trends in the coefficients.

For application to longleaf pine, we replaced coefficients that exhibited a trend with

crown ratio with a function of crown ratio. The inclusion of crown ratio as a continuous

variable improved by at least 16 percent the mean square residual for both models. The

authors’ model performed better on the modeling dataset based on fit statistics and on the

validation dataset. It also contained fewer parameters and was casicr to rearrange to solve

for height to a given diameter.

Keywords: Crown ratio, longleaf pine, Pinuspalustris Mill., tree taper equations, tree

taper model.
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