United States

Department of

Agriculture

Forest Service

on

New Orleans,

Louisiana

F&s;xv;h Paper

Functions fo

ting Product

Natural Shortleaf Pi

Robert M. Farrar, Jr., and Paul A. Murphy
SUMMARY

Taper (stem-profile) functions are presented for natural shortleaf pine (Pinus

e&in&u Mill.) trees growing in the West Gulf area, but they should also have

applications elsewhere. These functions, when integrated, permit the

prediction of volume between any two heights on a stem and, conversely by

iteration, the volume between any two diameters on a stem. The integrated

equations generally predict cubic-foot volumes that are within 2 1 cubic foot of

observed volume and account for at least 97 percent of the variation in volume.

Examples are given of use of the functions, and a computer program is

available to evaluate the functions for cubic-foot volume and compare them

with other volume equations.

ACKNOWLEDGMENTS

The authors are indebted to Mr. Daniel A. Yaussy, currently

located at the Northeastern Forest Experiment Station, Columbus, OH, for his conscientious assistance in gathering the field

data in southern Arkansas and northern Louisiana. We are further indebted to Dr. Edwin R. Lawson of the Southern Forest

Experiment Station, Oxford, MS, for supplying the field data from

western Arkansas.

May 1987
ERRATA June 1987

USDA, Forest Service, Southern Forest Experiment Station,

Research Paper SO-234, page 4.

Table 4 in R. P. SO-234 contains 3 erroneous values. The location

of these errors is denoted by the 3 underscored data values in

the reproduction of Table 4 below. These errors have been

corrected and the correct values appear in the reproduction of

Table 4 below.

Table 4. Coefficients for lower and upper stem taper functions by o.b., i.b.,

and 3 CR classes, natural shortleaf pines, 342 trees.

Equation n g1 g2 bl b2 b3 b4

----------------------------------------------------------------------------------------~

CRI, - l 13253541 25 l 385423 2.279039 -.044477 -23.637118

o.b.

CR2, -. 11988464 19.513315 1 .'i'72916 -.026344 -18.120387

o.b.

CR3, -.I1799179 4.995668 2.091531 -.027642 -10.484750

o.b. --------

CR+l, -.I2195134 ~372776 .936758 19.473495 2.066904 - l 038933 -17.738120

i.b.

CR2, -. 10905991 -.406013 .930204 13.933809 1.593209 -.020028 -12.575889

i.b. ----w-----

CR3,

i.b.

-.I1164159 -. 534799 .935277 -1.878825 1.954822 -.023757 -3.780569

---m----

The correct coefficients were included in the BASIC program

available to evaluate the taper functions. Therefore, Tables 5,

6, and 7 of R. P. SO-234 are correct.

Taper Functions for Predicting Prod

in Natural Shortleaf Pines

Robert M. Farrar, Jr., and Paul A. Murphy

INTRODUCTION METHODS

How would you like to have a prediction system

that estimates the tree volumes for almost any assortment of products imaginable? Foresters have long

pursued the notion of devising a system of tree stemvolume predictions that are both accurate and universal regarding merchantability specifications and

units of measure. Through the use of integrated stem

taper functions (also called stem-profile functions),

which assume that the tree stem is a solid of revolution, we are coming closer to this goal. Similar to D2H

td.b.h.” x total or merchantable height) volume functions, such taper functions could be slightly biased if

they are developed from one population of trees and

used to obtain estimates, perhaps, for a different population. However, any such resulting bias is usually

minor and often offset by the utility of these functions

in mensurational applications.

A total of 342 felled shortleaf tre ere measured

in a rectangular distribution of d. total height,

and crown-ratio classes in north Louisiana and south

and west Arkansas. These trees varied in d.b.h. from

1 to 26 inches, in total height from 8 to 107 feet, and

in crown ratio from 16 to 86 percent (tables l-3).

Taper functions can be regarded as infinitely variable D2H volume functions. Essentially, the diameters at any two heights or the heights to any two

diameters on the stem can be predicted, which in turn

can be converted into various units of volume or surface area. If properly constructed, they resolve the

intersection problems that can occur in the construction of a family of D2H volume functions. However,

taper functions are difficult to evaluate without considerable computing hardware, but their saving grace

is the variety of output they afford.

The selected trees were single-stemmed, reasonably

free of crook or sweep, and visibly undamaged. Samples were drawn from two areas: (1) south Arkansas

and north Louisiana (215 trees) and (2) west Arkansas (127 trees). In the first sample, stems were measured for d.o.b., bark thickness, and height at each

l-inch taper step from the stump to the zero-d.o.b. tip.

In the second sample, which had already been collected, stems were measured for d.o.b., bark thickness, and height to variable taper steps from the

stump to the zero-d.o.b. top. Also, in both samples,

d.o.b., bark thickness, and height were measured at

the stump, breast height, and the live-crown base on

each tree.

We used multiple regression (SAS 1979) to fit these

d.o.b., d.i.b., and height data to the stem-profile model

of Bennett and others (1978) to obtain predicted values of stem diameter given height of the diameter and

the tree d.b.h., total height, and crown-ratio class.

The model is:

This paper reports on a taper function model fitted

with data for naturally regenerated shortleaf pine

Pinus e&in&u Mill.) trees in the West Gulf area.

Since shortleaf pine has the widest range of the four

major southern pines, this information should be of

interest in the Central States and the Northeast and

also in the East Gulf area and the Southeast. About 96

percent of the shortleaf cubic-foot volume probably

occurs in the area served by the Southern and Southeastern Forest Experiment Stations (Sternitzke and

Nelson 19701, but outside this area the species can be

locally quite important. Kentucky and Missouri together probably contain over 75 percent of the northern shortleaf volume and West Virginia contains

about 10 percent.

D&/4.5)” if h, 5 h 5 4.5, (la)

d= or

I

D(H - h)/(H - 4.5)

+ bl(H - h)(h - 4.5YH2

+ b2D(H - h)(h - 4.5)/H2

I+ b3D2(H - h)(h - 4.5YH2

+ b*(H - h)(h - 4.5)

c(2H - h - 4.5)/H3 if 4.5 : h 5 H, (lb)

Farrar and Murphy are principal meneurationiete at Forestry Sciences Laboratory, Monticello, AR, Southern Forest Experiment Station,

Forest Service-USDA, in cooperation with the Department of Fore& Resources and the Arkansae Agricultural Experiment Station,

University of Arkaneaa at Monticello.

1
Table I.--Distribution of felled natural shortleaf sterns, CR < 36%

D.b.h. Total height class (ft)

class

(in) 10 20 30 40 50 60 70 80 90 100 110 120

.--

1 3 4

2 4 2 1

3 4 7 4 2

4 2 2 2

5 1 2

6 2 2 1

7 1 1

8 2

9 2 11

10 1 h 1

11 1 1 2

1 2 2

1 3 1 1 2

14 1 2

15 1 1

16 1

1 7 2

1 8 1

19 1

2 0

2 1 1

2 2

2 3

2 4

2 5

2 6

2 7

2 8

2 9

3 0

3 1

3 2

3 3

3 4 Total = 77

=:

= meter at brea

ZZ

E

for the base of the

stem (h 2 4.5 feet) and upper stem (h E- 4.5

feet), but the overall model is constrained such that

the two stem sections are joined at breast heigh

These expressions were fitted with both outside- a

inside-bark data and for eat of three ~ro~~-ratio

(CR) classes ,which were

2

Cubic-foot volumes fro the integrated taper function (solid of ~ev~~~tio~)

i

u d2db,

hl

(3)

where

diameter at height h from

any of the four taper sections implied

equations Pa and lb, for a selected

class,

h, = an upper height, and

hl = a lower height.

The solutions of eq ation (3) are not presented

herein because of their length and complexity but are

available in general form in the appendix of the paper
Table %-Distribution of felled nc~tural shortlaafstems, G

D.b.h. Total height class (fi)

class

(in) 10 20 3 0 40 5 0 6 0 70 80 90 100 110 120

1

2

3

4

5

6

7

8

9

PO

1 1

12

1 3

1 4

15

16

17

1.8

19

2 0

21

2 2

2 3

2 4

2 5

2 6

2 7

2 8

2 9

3 0

3 1

3 2

3 3

3 4

5 1

2 3 2

3 2 3

2 4 3 2

1 1 1

3 5 1

3 6 1

2 2 2

5

5 5

1 5

4 3

1

4

3

1

I

1

1

2

2 2

1 1 P

1 3

1 2

2 11

1 1 1

P 2

1 2 1

1 1

1

1

2 2

1 1

1

Total = 141

by Bennett and others (1978). They are specifically

available from the authors in a

evaluate the integrated shortleaf

for cubic-foot volumes (see Appendix).

We also tried the mod of Matney and Sullivan

(1980), but the model of ennett and others (1978)

ed as well and was fitted more efficiently.

ermine “goodness of fit ,“observed cubic-foot

alues were calculated per tree for the 342

trees by assuming conic sections between ~o~secutive

specified taper steps

compared with corre

gram was written in

uterr to produce the predicted

renouncement box in the

program also allows

umes via other av ations for comparison purposes

d.b.h. and total h

ention of co~~erci~~ product names does

by the USDA Forest Service nor exclude

ucts that may serve as well.

enciorselar prod-

~Qe~cierlts for the lower and the upper stem funeLions by o.b., Lb., and three crown-ratio classes are

‘ven in table 4. Note the consistent trends in the

~oe~~ients with crown-ratio class. This implies that

could be incorporated into the functions as a conNo conceptual problems prevent

,and Vale&i and Cao (1986) present

n&ions employing this

a discrete variable.

profiles (d.o.b. by

re given in figure 1.

classes is shown for
Table 3.-Distribution of felled natural shortleaf stems, CR > 50%

D.b.h. Total height class (ft)

class

(in) 10 2 0 3 0 4 0 5 0 6 0 7 0 8 0 9 0 100 110 120

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

1 6

1 7

18

19

2 0

2 1

2 2

2 3

2 4

2 5

2 6

2 7

2 8

2 9

3 0

3 1

3 2

3 3

3 4

7 2

3 4

3 3

2 3

2 1

2

2

1

1

1

1

3

4

3

2

2

1

1

1

2

1

1

1

2

2

2

4

3

1

3

1

1

1

1

1

2

1

2

1

Total = 124

Table 4.-Coefficients for lower and upper stem taper functions by o.b., Lb., and 3 CR classes,

natural shortleaf pines, 342 trees

-

Equation n g1 lT2 bl bz b3 b4

CRl, -.13253541 25.385423 2.279039 -.044477 -23.637118

o.b.

CR2,

o.b.

-.11988464 19.513315 1.772916 -.026344 -18.120387

CR3,

o.b.

-.11799179 4.995668 2.0915314 -.027642 -10.484750

CRl,

Lb.

-.12195134 -- .372776 .936758 19.473495 2.066904 -.038933 -17.738120

cm

i.b.

-.10905991 -.406013 .930204 13.933809 1.593209 -.020028 -11.923614

CR3, -.11164159 -534799 .935277 -1.878825 1.9548223 -.023757 -3.780569

Lb.
(2) D.b.h. = 8 inches, total height = 50 feet,

CR = same classes as in (l), and

(3) D.b.h. = 4 inches, total height = 30 feet,

CR = same classes as in (1).

Note in figure 1 that a higher CR class (percent)

generally suggests lower stem volume due to greater

taper in the long crown section, and this is generally

borne out in evaluations of the integrated function for

volumes (see APPLICATIONS section). However, figure 1 also shows that even though a high CR implies

low volume, in certain lower stem sections of a high

CR tree the predicted taper is less than that of a similar low CR tree and, hence, a greater volume is suggested for these sections. These relationships were

also noted by Dell (1979), Farrar (1987), and Farrar

and Murphy (1987).

The “goodness of fit” statistics in tables 5 and 6

show how well the taper functions reproduced the observed o.b. and i.b. cubic-foot stem volumes. Definitions pertaining to these tables are:

n = number of observations,

Ti = mean observed (0) value,

a = mean difference between predicted (p)

and observed

%a

= zp - 0)/n,

= percent mean difference

= [E:((p - o)/o)/n1(100),

RMSd = root mean squared difference

= X&p - o)2/n,

RMS%d = root mean squared percent difference

= w)/o)2/n (loo),

FI = fit index

= 1 - [X(p - o)s/z(o - $21,

Total stem cubic feet = volume of a stem from a

O.l-foot stump to the bud tip for trees with

d.b.h. >O inch,

Merchantable stem cubic feet = volume of a stem

above a 0.5-foot stump to a 3-inch d.o.b. top for

trees with d.b.h. >3.5 inches, and

Sawtimber stem cubic feet = volume of a stem

above a 0.5foot stump to an &inch top d.o.b. for

trees with d.b.h. >9.5 inches.

DISCUSSION

The fit statistics indicate that the taper functions

adequately represent the stem volumes for shortleaf

pine. Mean differences (d) are generally less than 1

cubic foot, indicating a relatively low positive bias.

Root mean squared differences (RMSd) are generally

less than -t-6 cubic feet, indicating good precision. The

fit indices (FI), which represent the amount of volume

variation accounted for by the taper functions, are all

97 percent or better, also indicating a good fit.

In general, the shortleaf pine taper functions remap

the observed volumes slightly better than recently

developed functions for natural loblolly pines (Farrar

and Murphy 1937) by all criteria except fit index (FI),

which is consistently just slightly better for loblolly.

The stem taper in shortleaf, particularly in the crown,

appears more regular and smooth than that of loblolly

and may account for the shortleaf advantage. The better fit indices for loblolly are probably due to the fact

that the loblolly trees were larger, and similar deviations produced a better fit index for larger volumes

because the volume variances were also larger.

0.5 5 IO I5 2 0 25 30 3 5 4 0 4 5 5 0 5 5 6 0 6 5 70

HEIGHT (FEET,

Figure l.-Predictedd.o.6. by height and CR for three example shortleaf pine stems.

5
Table K-Fit statistics (outside bark) fOF nntuml shortk?af pine taper equatiorw,

342 trees measured at 1 -inch and varying taper steps

Equation n 0 ii ?a RMSd RhIS%d FI

ft3 p % ft3 %

Total stem cubic feet, o.b.

CRl, o.b. 77 18.366 .311 - 3.550 2.565 14.685 .9912

CR2, o.b. 141 28.413 .698 - .153 3.978 10.947 .9870

CR3, o.b. 124 33.270 .591 - 6.104 4.089 17.973 .9892

Overall 342 27.912 .572 - 3.075 3.751 14.638 .9888

Merchantable stem cubic feet, o.b.

CRl, o.b. 4 6 29.409 .390 - 5.886 3.272 15.090 .9876

CR2, o.b. 120 32.257 .701 - 1.930 4.287 12.399 .9849

CR.3, o.b. 100 39.979 .553 - 1.127 4.521 11.911 .9867

Overall 266 34.668 .591 - 2.312 4.222 12.731 .9862

Sawtimber stem cubic feet, ob

CRl, o.b. 2 4 45.865 1.019 1.523 4.673 11.716 .9731

CR2, o.b. 75 45.174 .710 - .634 5.572 12.667 .9763

CR3, o.b. 7 0 52.485 .311 .658 5.591 11.167 .9790

Overall 169 48.300 .589 .207 5.462 11.931 .9775

Table 6.-Fit statistics (inside bark) for natural shortkafpine taper equations, 342

trees measured at 1 -inch and varying taper steps

Equation n 0 ii o/ad RMSd RMS%d FI

ft3 fli? % ft3 %

Total stem cubic feet, Lb.

CRl, i.b. 7 7 15.510 .231 - 4.590 2.470 17.574 .9892

CR2, i.b. 141 23.890 .523 - .9!34 3.659 17.312 .9854

CR3, i.b. 124 28.031 .437 - 7.523 4.102 29.490 .9857

Overall 342 23.504 .426 - 4.171 3.605 22.548 .9864

Merchantable stem cubic feet, i.b.

CRl, i.b. 4 6 24.988 .280 -4.215 3.175 13.932 .9848

CR2, i.b. 120 27.216 .539 .145 3.949 14.608 .9832

CR& i.b. 100 33.777 .423 2.543 4.545 16.001 .9826

Overall 266 29.297

Sawtimber stem cubic feet, i.b.

.450 .293 4.067 15.036 .9833

CRl, i.b. 2 4 39.501 .585 1.539 4.517 12.653 .9679

CR2, Lb. 7 5 38.433 .456 .177 5.095 15.033 .9745

CR3, i.b. 7 0 44.710 .121 2.317 5.573 15.401 .9734

Overall 169 41.185 .335 1.257 5.223 14.875 .9736
oth the loblolly and shortleaf fit statistics are

tly poorer than those reported by Farrar (1987)

tural longleaf stems, where the mean difference

nerally less than CO. 1 cubic foot, the RMSd less

+3 cubic feet, and the fit index greater than 98

nt. This may be due to longleaf stems being more

arly formed with smoother taper and easily

ed live crown bases. The stem taper in loblolly

crown sections is often quite irregular and approaches

a step-function in shape. Shortleaf stem taper appears

reasonably regular and smooth, but the base of the

live crown is often difficult to locate precisely because

live limb size and vigor often decrease gradually down

the stem, with no sharp demarcation for a crown base.

APPLICATIONS

edicted cubic-foot volumes for the example trees

in figure 1 are given in table 7. The volume definitions

are the same as those in tables 5 and 6 except that, for

convenience, a O&foot stump is assumed throughout,

as in figure 1. Note that, as suggested, a high CR

ge implies a relatively low stem volume. These

Pr volumes are output from evaluation programs written in BASIC for an IBM-PC XT microcomputer. See the associated Software Announcement

box for reference and source of this program.

One thing is noteworthy in table 7. For the 12-inch

7Q-foot tree, shortleaf volumes for CR3 (high crown

ratio) are slightly greater than those for CR2

(medium crown ratio). This bears out the earlier

statement that a larger predicted volume is possible

terns of high crown-ratio trees that have

orer upper stem form.

s of the predicted values in table 7 with

similar predictions for loblolly trees (Farrar and Mur1987) and longleaf trees (Farrar 1987) indicate

for the same d.b.h., total height, and CR class the

shortleaf values are generally somewhat larger than

those for loblolly and often somewhat larger than

those for longleaf. For this study at least, this would

imply that shortleaf generally has better form than

loblolly and sometimes has better implied form per

CR class than longleaf.

Board-foot volume predictions are also possible by

writing a program to utilize equations (la) and (lb)

and a formula log rule such as Doyle or International

l/4-inch. A diagram log rule can also be used by storing its array of values by scaling diameter and log

lengths or, better, by using a functional fit of the diagram rule.

To obtain predicted board-foot volumes, equations

(la) and (lb) would first be evaluated to obtain scaling

diameters at the top of each simulated log (or fractional log). Then the scaling diameter ar.d log lengths

could be passed to the log rule function ‘;o obtain predicted board feet per log. Finally, the board-foot volumes per log could be summed to obtain predicted

stem volume. Note that equations (la) and (lb) are

evaluated using the cumulative height to the top of

each log (or fraction), including stump height and

trim allowances. For example, the height of the first

log might be 17.3 feet, which would include a l-foot

stump, a 16-foot log, and a 0.3-foot trim allowance.

The height to the second log might be 33.6 feet

(17.3 + 16.3). An actual evaluation program such as

this is beyond the scope of this paper.

If cubic-foot volume equations in terms of d.b.h. and

total height are available for a locale, the adequacy of

the taper functions for that locale can be easily

checked via the available BASIC evaluation program.

Either inside- or outside-bark equation values for

total volume or for volume to specified top diameters

can be compared with those predicted by the taper

functions for the d.b.h.-height combinations involved

in the volume equations. The program could also be

modified to predict volumes for merchantable heights

Table I.-Predicted cubic-foot volumes for the three example shortleaf stems in figure 1

Merchantable Sawtimber

Total CR Total volume volume volume

D.b.b. height class o.b. i.b. o.b. i.b. o.b. i.b.

in Tt -___-___--____-_-_ _ _---__--- ft3-_-___---____--____---------

12 IQ 1 29.584 24.784 29.469 24.696 25.585 21.546

2 28.408 23.495 28.210 23.389 24.127 20.036

3 28.644 23.596 28.498 23.489 24.378 20.231

8 5 0 1 9.252 7.504 9.122 7.407 n.a. n.a.

2 8.869 7.081 8.721 6.972 n.a. n.a.

3 8.452 6.610 8.261 6.484 n.a. n.a.

4 30 1 1.347 .981 .730 ,523 n.a. n.a.

2 1.308 .929 .774 539 n.a. n.a.

3 1.077 .691 .684 .444 n.a. n.a.

~-

n.a. = not applicable.

7
and thereby aceomodate comparisons with usersupplied equations that are in terms 0fd.b.h. and merchantable height.

CONCLUSION

In the Midsouth, we now have six integrated taper

(stem profile) functions for southern pines that employ the same basic model (Bennett and others 1978)

and use the same crown ratio classes as a surrogate

for stem form. These functions are:

Planted slash pine-Dell and others (19’79)

Planted loblolly pine--Fedueeia and others (1979)

Planted loagleaf pine-Baldwin and Polmer (1981)

Natural lolrgleaf pine-Farrar (1987)

Natural shortleaf pine-reported herein

Natural lot~!olly pine-Farrar and Murphy (1987)

This is not t&s only available model and others may

serve as well, but this model is reasonable in form,

easy to fit, and highly serviceable, as afirmed by the

above use. The commonality of functional form aliows, 1 ithin reason, direct comparisons between species’ profile functions. The most obviously needed improvement would include crown ratio as a continuous,

rather than a discrete, variable and thereby improve

the utility. However, the present three CR classes

seem to be adequate as discriminatory devices for

many purposes. They are not unique but were simply

the most logical groupings found in the planted slash

ine data base (Dell and others 1979) and have been

sed since in the other species’ functions to maintain

~~~~iste~~y"

LI ATURE CITED

aldwin, V. C.; Polmer, B. H. 1981. Taper functions

for unthinned longleaf pine plantations on cutover

est Gulf sites. In: Proceedings, 1st biennial southern silvicultural research conference; 1980 November 6-7; Atlanta, GA. Gen. Tech. Rep. SO-34. Washington, DC: U.S. Department of Agriculture, Forest

Service: 156-163.

ennett, F. A.; Lloyd, F. T.; Swindel, B. F; Whitehorn,

E. W. 1978. Yields of veneer and associated products from unthinned, old-field plantations of slash

pine in e north Florida and south Georgia flatwoods. es. Pap. SE-176. Ashville, NC: U.S. Department of Agriculture, Forest Service, Southeastern Forest Experiment Station.. 80 p.

Dell, T. R. 1979. Potential for using crown ratio in

predicting product yield. In: Proceedings, forest resource inventories workshop; 1979 July [days of

month unknown]; Ft. Collins, CO: Department of

Forest and Wood Sciences, Colorado State University; 1979: 843-851. Vol. II.

Deli, T. R.;Feduccia, D. P.; Campbell, T. E. [and others]. 1979. Yields of unthinned slash pine plantations on cutover sites in the West Gulf region. Res.

Pap. -147. New Orleans, LA: U.S. Department of

Agri ure, Forest Service, Southern Forest Experiment Station. 84 p.

Farrar, R. M., Jr. 1985. Crown r.atio used as a surrogate for form in a volume equation for natural longleaf pine stems. In: Proceedings, 3rd biennial southern silvicultural research conference; 1984

November 7-8; Atlanta, GA. Gen. Tech. Rep. SO54. Washington, DC: U.S. Department of Agriculture, Forest Service: 429-435.

Farrar, R. M., Jr. 1987. Stem-profile ftmctions for predicting multiple product volumes in natural longleaf pines. Southern Journal of Applied Forestry (in

press).

Farrar, R. M., Jr.; Murphy, P. A. 1987. A versatile

volume-defining function for natural loblolly pines.

Res. Pap. SO--. New Orleans, LA: U.S. Department of Agriculture, Forest Service, Southern

Forest Experiment Station (in process).

Feduccia, D. P.; Dell, T. R.; Mann, W. F., Jr. [and

others]. 1979. Yields of unthinned loblolly pine

plantations on cutover sites in the West Gulf region. Res. Pap. SO-148. New Orleans, LA: U.S. Department of Agriculture, Forest Service, Southern

Forest Experiment Station. 88 p.

atney, T. G.; Sullivan, A. D. 1980. Inside and outside bark taper equations for old-field plantation

loblolly pine trees in Central Gulf Coastal plain.

Tech. Bull. 101. Starkville, MS: Mississippi Agricultural and Forestry Experiment Station. 28 p.

Statistical Analysis Systems Institute. 1979. SAS

User’s Guide. Cary, NC: Statistical Analysis Systems Institute; 494 p.

Sternitzke, H. S.; Nelson, T. C. 1970. The southern

pines of the United States. Economic Botany. 24(2):

142-150.

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

improve loblolly pine taper equations. Canadian

Journal of Forest Research. 16: 1141-1145.
ppendix

SOFTWARE ANNOUNCEMENT

Upon request, a printout of a BASIC program to

evaluate the natural shortleaf pine stem-profile function discussed in the accompanying article is available from the authors. Inputs are tree d.b.h., total

height, crown-ratio class, stump height, and top diameter. Outputs are predicted cubic-foot volumes, i.b.

and o.b., from the profile function; “observed” volumes

from user-supplied volume functions; and basic

“goodness of fit” values generated from a comparison

of predicted and “observed” values. The example trees

from the article are evaluated as examples and test

problems. The program was written for the IBM-PC

XT microcomputer but sb.ould be easily adapted to

other BASIC dialects for use with other machines.

aLJ.S. COV!ZRNMENTPRINTlNCOFFlCEl987 -734 .029/ 40 63 3 REGIONNO. 4
Farrar, Robert M., Jr.; Murphy, Paul A. Taper functions for

predicting product volumes in natural shortleaf pines. Res.

Pap. SO-234. New Orleans, LA: U.S. Department of Agriculture, Forest Service, Southern Forest Experiment Station;

1987. 9 p.

Taper (stem-profile) functions are presented for natural

shortleaf pine (Pinus echinatu Mill.) trees growing in the

West Gulf area. These functions, when integrated, permit

the prediction of volume between any two heights on a stem

and, conversely by iteration, the volume between any two

diameters on a stem. Examples are given of use of the functions, and a computer program is available to evaluate the

functions for cubic-foot volume equations.

Additional keywords: Naturally regenerated shortleaf

pine, predicting product volumes, taper function model.