Un~ted States Taper and Volume Equations 

Forest Service for Selected Appalachian 

Research 

Paper NE-490 Hardwood Species 

1981 

@EST Sf- @ %.rirsrw by A. Jeff Martin 
A. JEFF MARTIN attended Miehigm Stab University, receiving a degree in forestry in 1965, an M.S, degree in forest 

mmagement in 1966, and a Ph,D de~ee in forest management in 1969. Me joined the staff of the Northeastern Forest 

Experrimen* Station at the Forestugr Sciences Labomtoq, 

PGnceton, \Vest Virginia, in 1969. He is eunentfy with the 

Northeastern Station's project on increawd supply of hadwood raw rnate~ds, 

MANUSGRPT RECEIVED FOR PUBLICATTON 

17 FEBRUARY 1981 

Coefficients for five ta,ger/volume models are developed for 

I8 Appdachizrll hardwood species. Each model can be used 

to estimate &meter ad any point on the bole, height to any 

preselected diameter, and cubic-1Foot volume between any 

two points on the bole. The resulting equations were tested 

on six sets of independent data and an evaluation of these 

tests is included, A wide variety of volume tables em be constmeted with the models; some exmgles are given. 

Table sf Gonknts 

Data Collection. ............................ .I 

hdysis ................................... 3 

Testing the Equations. ........................ -6 

Discussion and Application ..................... .9 

Liknltuue Cited ........................... .I1 

Appendix A. ........................... .I2 

Appendix B ............................ .I6 

Appendix C ............................ .I8 
Errata Sheet 

For Research Paper NE-490 the following corrections should be made: 

H = height up the bole from groundline to lower should read 

H height up the bole from groundline (feet) 

Polynminal should read polynomials 

The equation should read 

D' /DBH' - (idbo) (DBH'~' -' 
Gument interest in multiple-product tintzbrsr Inawesting hm 

genemted a need for improved volumes pre&elion for in&- 

vicfud t~es and yield pre&c~on for stands. Knorivledge of 

tot& cubic-hot or board-foot volumes is no longer suE6cient. 

We now need to how what gadions of a tree ean be used 

for speeisc products, and we need do identi@ the en~re 

may of produets that can be obtained from specific stsmds, 

When avdable, empirical d&a -81 provide some of the 

answers; however, such infom,lition will generally be restl.lickd to existing product specifications, merefore, to 

ink~olate iuld extmpolate eempiried data, and to provide 

greakr aexibi1it;Jr with chan@ag specifications ;and with new 

produets, a more generalized approach is needed, One of the 

most important etements in such a syskm is reliable taper1 

volume equations for the species in question. Such equations 

enable the user to estimak the diameter ad any point on the 

bole, the height to my predetermined diameter, ad the 

volume between any two points on the bole, 

Althcrugh vadous methods of dcrrrefoping taper/volurne equations have ken proposed (~ozik and others 1969; Kozak 

and Smith 1966; Mm md Burkhart 1976; Fries and Matem 

1965; Bennett and Swbdef 1972; Gouiding and knluuray 

1976; Demaeusekalk f971,1972,1973a, 1973b; Bmee and 

otheus 1968; Omerod 1973; Cfaatter 1980; Gao and others 

1980; and Demaerschdk iand Kozak 1977), the infomation 

is either theoretical or limited puimari;ly to softvvood species. 

Until recently taper hnctions for eatern hardwood species 

were not available. However, the work by Wilt (1980) for 

upland oaks in the Midwest, coupled with the results repoYted herein, should fill most of the voids. 

Dada CoUle~tion 

The objective of this studlJr wm to develop taper and volume 

equations for selected cornrnevcjlal Appalachian hadwood 

species, Stem measurements were collected from 1,162 trees 

in West Virginia and southweskm Virginia, Eighken species 

we= sampled and data were obtained from 39 different 

stands on poor to exeeltent sites. Data for Black tupelo md 

black oak cme from only 5 stands, whemrjlm sugar rnlsiple 

measurements erne from 17 locations; the average was 10 

stmds per species, 

nees without forks below midkeight were randomly selected, dth stratidcation by species and dbR (dimehr at 

breast height) class. Although a balanced dist;pjibution by dbh 

and total height was desi~d, dbh alone ws used in selecting 

sample trees for practicafiGy md economy. me resulting 

smple ws well bafmeed by dimeter elass for nearly all 

species, whereas the totd height distri;butiorr was somewhat 

uneven. With velry few exeept;ions, thme to five trees per Iinch &meter efas {between 5 and 22 inches) were selected 

for each species, 
No. of 

Species Vaviable Range Mean SD samp te 

trees 

Red maple dbh (inches) 5.1 - 22.0 12.4 5 -0 

total height (feet) 48.1 - 103.6 76 -2 12.7 76 

Sugar maple dbh 

total height 

Sweet birch dbh 

total height 

Yellow birch dbh 

total height 

Hickory, sp. dbh 

total height 

American beech dbh 

total height 

White ash 

Yellow-poplar 

dbh 

total height 

dbh 

total height 

Cucumbertree 

Black tupelo 

dbh 

total height 

dbh 

total height 

Black cherry dbh 

total height 

White oak dbh 

total height 

Scarlet oak dbh 

total height 

Chestnut oak dbh 

total height 

Bed oak dbh 

toM height 

Black oak dbh 

total height 

Blaek locust dbh 

total height 

herican basswood dbh 

total height 

All species dbh 

total height 

Dbh of the smpfe trees (Table 1) ranged from 4.6 to 22.5 Dbh and bark thickness (at breast height) we= measured and 

inches mean = 12.8 inches) and total height ranged from recorded for each sample tree. Stem prof"r1e data (diametem 

35.1 to 118.5 feet (Mean = 77.1 feet), Sample size ranged outside bark and height above ground) were obtained at eight 

from 45 trees for black oak to &;I trees for white oak. points on the bole: stump (I foot above ground) and at ap- 
proximately 1/8,1/4,3/8,1/2, 5/8,3/4, and 7/8 of total 

height. Stem measurements for most of the sample trees were 

obtained during the fall, winter, and spring with a Barr and 

Stroudl optical dendrometer; however, the sample also includes direct measurements from 246 felled trees. Total 

height (in feet, from groundline to tip) was obtained either 

with the dendrometer or from direct measurement of felled 

trees. 

A variety of published taper equations was examined and 

tested on the stem measurements in the sample. The models 

ranged from simple to very complex; from models with 

only one coefficient to semented polynominds having two 

join points and six coefficients. Some were discarded because 

they did not fit the data well and prediction was poor; others 

were discarded because they were too complex. 

The five models that were retained (Bruce and others 1968; 

Demaerschalk 1972; Kozak and others 1969; Max and 

Burkhart 1976; and Ormerod 1973) still ran the garnut from 

simple to complex (Tables 2 - 6); however, they all fit the 

data reasonably well and yielded good predictions in subsequent tests. Variables and regvession coefficients common 

to all five models are defined as follows: 

DBH = diameter at breast height (inches) 

TH = total tree height from groundline to tip 

(feet) 

H = height up the bole from groundline to lower 

D - dimeter at height "H" (inches) 

HL = height up the bole from groundline to lower 

limit of volume calculation (feet) 

HU = height up the bole from groundline to 

upper limit of volume calculation (feet) 

V = volume of bolewood section between 

"HL" and "HU * (cubic feet) 

bo - b6 = regression coefficients estimated from 

the sample data. 

In Tables 2 to 6, equation (I) estimates bole diameter at 

any height H above ground. Tkis form of each model was 

used in all andyses to estimate the regvession coefficients 

for the sample data. Equation (2) is the inverse of equation 

(1); it estimates the height above ground to any preselected 

bole diameter D. Equation (3) estimates cubic foot volume 

The use of trade, firm, or corporation names in this publication is for the information and convenience of the reader. 

Such use does not constitute an officid endorsement or approval by the U.S. Department of Agriculture or the Fomst 

Service of any product or service to the exclusion of others 

that may be suitable. 

Table 2.-Bruce and others' taper and volume functions 

D2 /DBH2 = bl (XI5 ) + b2 (XIS - x3 )(DBH)(I~-~ ) 

+ b3 - X3 )(TH)(IO-~ ) 

+ b4 (xlS - x~~ )(TH)(DBH)(~~-5 ) 

+ bg (xiS - x32 )(TH*~ ) 

4 b6(pS - X~~)(TH~)(~O-~) (1) 

Where: 

X = (TH - H)/(TH - 4.5) 

H : Must use an iterative method; changing the value 

for H in equation (1) until the predicted value of 

D is satisfactorily close to the desired value for D (2) 

Where: 

XU = (TH - HU)/(TH - 4.5) 

XL = (TH - HL)/(TH - 4.5) 

A = (b, )(lo-' ) + (b2)(~~~)(10-2 ) + (b3)(~~)(10-3 ) 

+ (b4)(~~)(~~~)(1~-5 ) + (b5 )(THOa5 )(lom3) 

+ (b6 )(TH~ )(lom6 ) 

B = - Eb2 )(DBH)(~O-~ ) + (b3 )(TH)(~O-~~ 

c =- ~b4)(~~)(~~~)(1~-5 ) + (b5 )(THO' )(Io-~I] 

E = - kb6)(~~2)(10-6$ 

of the bole between HL and HU. Equation (3) was derived 

by integrating equation (1) between the limits HL and HU: 

The model by Bruce and others is a rather lengthy polynominal that the authors refer to as their "final equation" in 
Table 5.--...Max and Burkhaul" taper and valume 

lunc2ions 

TabEe 3.----DemaerschaIk's taper and vslume functions 

Where: 

XI: =; TH- HL 

Table BO,-Kora&r and 08hers""taper and volume 

functions 

+ bj (a, - HITH)~ I, + b, (a2 - H/TH~' 12 (11 

Where: 

a,, az = join points 

Il = I, H/TH G a, 

I2 - 1, HjTE G a' 

- 0, H/TH > a2 

Where : 

A = b2 + I; b3 + I: b4 

dl = estimakd dimeter at height al TH 

d, - estimated diarneter at height az TH 
Table 6.-Ormersdb taper and valurme furscllorrs 

V = (0.005454 DBH' ) (4.5 - TH) 

"id --I 

Where: 

the development of red alder taper and volume sysdems. The 

model is conditioned so that D = 0 when H = TH. En addition 

to its size, particulatsly in the expression for volume, the 

model has mother dissrdvantage for some users: It emnot 

be rewdtten in terms of H; therefore, estimating height 

above gmund to the &inch mmk (for exmple) must be 

handled by an iterative procedure (e,g., inbrval hdving, 

Mewton-Raphson method, ete,) 

Mm and Burkbau"t% model is also a rather complicated approach to taperlvolume equations. Their approach was to 

develop three separate submodels that desc~be the rreiloid 

fmstum of the lower bole, the pmaboloid fmsGum of the 

middle bole, md the eonied shape of the upper portion, The 

thrcee submodels are then spliced together at two ""join 

points" into an overdl sewenled polynominal tree model, 

The version selected for this study was their quadraticquadratic-quadratic model. The equation is conditioned so 

that D = 0 when H - TH, To use this approwh, one must 

first decide which of the thueo bole segrnents is approg~ate; 

this deternines the values for the two dummy vauiables. During the malyses, optimd join points were simultaneously 

estirnabd for each species dong with the reg~ession coefficients (Appendix Table 15). Join points are simply proportions of total height; the lower join point is usually close 

to 0.1 and the upper is usudly between 0.6 and 0-7. 

Both Demaemehalk's equation and the equation developed 

by Kozak and othevs are much easier to use, Both are conditioned so that E) = O when I.I = TH, and both em be rewdtten in Lems of H, The model by Kozak and others em 

easily be solved with any electronic cdculator possessing a 

swwe root function. Demaersch&utk9s model requires a somewhat more sophisticated machine sinee the exponents (re- 

~ession coefficients) a= nod whole numbers. hmaersekdk's 

taper equation ww depicved fmm the bade logathmie 

volume equation: 

Log V = a -+ b Log DEW +- c Log TW 

Therejfore, the volume equation obtkned by inkgrating the 

taper function (Equation 3, Table 3) is compatible with this 

o~gind logarithmic volume equation; that is, they botb yield 

the sme results. Actually, Demaersehdk's equation is compatible only if the model is not fitked separately for bper 

and volume. 

Omerodk equation has only one coefeeient that needs to be 

estimated, hence the basic taper equa.tjioa is verfr emy to us, 

The model is conditioned so that 13 - 0 when H = TH; and 6) = 

BBM when W = 4 '5 feet. 

All of the models were fitted to the sample data using a computer progrm for nonlinear reguession. mom were eight obervations per sample tree and each crbseniation consisted of DBH, 

TN, D, and W. me dependent va&able in each ease was D2 j 

DBH2 ; where D = dimetev outside bark and DBH = dbh outside 

bark. 

Results of dl the repession analyses are plresenkd in the Appendix (Tables 12 - 16). Including dl of the results allows the user 

to select whatever model he deems most appropriate. He can 

choose coefEcients for individual species or use those for all 

species combined. TE.lus a user has mat Elexibility in eustomizing a taper/volume prediction system to suit his needs. 

A summary of the regresion results is presented in Table 7. One 

can readily see that the= is little difference between models in 

how well they fit the sample data. Not too suuplPisingly, 

Omerod's model with only one coefficient does not perfom 

quite as well as do the mom complicated equations. However, 

Table 7.----Comparison of taper models for the overall 

equations (a1I species cambined); V = D2/08H2 

for all models a 

Model RZ Stmdard emor 

of the estimate 

Bmce and others .9 I2 

Dernaersehalk -863 

Kozak and others .864 

Max md Burkhart .9 11. 

Omerod ,800 

= dimeter oubide bark 

DBN = dimeter at brcsast height outside bark 
when one considem model eomplexiGy and limitations, as well 

as goodnetrs of fit as evidenced by the statistics in Table 7, the 

decision as to wkieh model is ""best" is anything but elear cut. 

Table $.---Summary sftatistics for the trees and 

pieces from the six tesl sites 

Test 

1 

Test Test Test Test Test 

2 3 4 5 6 

me equations were evaluated to see how well a taper-baed 

approach would work on independent sets of dab, and also to 

sc?e if any of the models were sigrrifiearrtly better than the 

othea, me tests compared dimeter prediction, height predietion, and the prediction of volume for difg~nt-sized pieees 

Gags, bolts, and entire mmerchmtable boles). 

No. of tmes 83 

Dbh (in): 

Mean 8.3 

Std, deviation 2.0 

Minimum 5.0 

Maximum 13 .0 

Tot& height (ft): 

me data used in the testing were obtained fmm six difhrent Mean 61 70 67 69 73 7'2 

stands (three in northern West Virginia, two in the middle of the Std, &viation 8 13 9 12 13 12 

state, and one in the southern portion), Species composition Minimum 42 25 44 46 41 48 

vafied farom Allegheny hadiwoods to upfmd hardwoods, Test Maximum 74 911 92 92 93 104 

area 1 ww predominahlg black cheruy, md test 6 was nearfy. No. of pieces kdf yelIow-poplar. Mean stmd age raged from 55 to 70 years, 

123 149 182 228 200 206 

and composite site index (for all species combined) varied from Piece length (ft): 

60 to 80 feet at 50 years. Mem 12.3 13,1 11.7 13.6 1. 14-9 

Std. deviation 12.9 13.4 -i 13.6 10.1 15.6 18.5 

me test data consisted of 1,088 pieces from 593 trees (T~ble 

8). Mean dbh of the test trees was 8.1 inches (rmge = 4.8 to 

19.1 inches) glpd mean total height wtsts 69 feed (range = 25 to 

104 IFeet). &an piece feag9.r was 13 feet; hawever, the size 

rmged from less than 1 hot for srrrdall cull sections to over 

82 feet for entire merchantable portions sf tho bale, 

Aetuai heights, to the lower and upper ends of each piece, wem 

used in eaeb equation to predict the lasge end and small end 

dimebrs outside bark. me p~dickdimeters were then eom 

pamd do the aeGrraI values using the frsflowring three critePia 

(eldapkd fmm Gao and others 1980): (I) tsisr; (the mean sf the 

differences between the predicted and achd vdues), (2) mean 

absoluk diffe~nce (the mean of the absoluk diEerenees), and 

(3) stmdaud deviation of the diffemnees. me sme proeedues 

we= followed for eompa~rrg height predictions; acbrud largemd smallend dimetercs wem usd to estimate lower and upper 

heights respectively. 

The esmpMson of volumes for each piece f'ollovved a similar 

pattern, except that no actual B~"z$uues were avairlable (i.e,, the 

pieces were not imme~ed in wader to debmine tme vs%urnes), 

Henee, volume compauiison vva between MO estimabs: 

Smdliiaazb fornula, vemus the volume equagons obtained by integrating the taper Ifunetisns. Volume (including bark) had been 

p~viously compukd by SmP;lim9s fornula for each log and bolt 

that vrrm bucked &om fie- test twes, Smai3im9s vlslZurrre far longer 

test pieces vvm detemiaed simply by summing the individual 

log and "aolt volumes contziiiuaed within the piece. 

All cornpadsons were milde by lasing the apgropdate eciefBcients for individud species in each model, Bi~es, mean abss8uk differences, md stmdard deviations for these eompdsons are shown in Table 9, 

Minimum 1,O 1.2 12 1.0 06 1-1 

Mrtxirvtunn 54-8 6$*2 611.5 72.2 72.6 82.5 

Dsb small 

end (in): 

Mean 6.3 6.7 5.3 6.0 6-6 5.5 

Std. deviation 1.8 2.8 1-4 23 2.6 1-65 

Minimum 3.7 4.0 4.1) 4.0 3-9 4.0 

Mmimum 10.6 15.5 11.0 15.8 16.0 12.1 

Dob large 

end fin): 

Mean 7-8 8.3 6.4 7.8 8.0 7.4 

Std. deviation 2.2 3.2 1.7 3 2 2 9 2.8 

Minimum $2 42 4.2 4.1 4-1 4.0 

Maximurn 13.8 18.7 112.7 19.2 202 21-1 

mese measures of =curacy and preeisicm indieate that all 

models do quite we11 in predicting diameter, height, and volume. 

Gomghng models in Table 9 is ewiejt. if we combined the pieces 

from all six sites and mconrpute the tfilcee test edbda (i.e., a 

welighted average of the vdues in Table 9). If we then imore 

any resulting negdive sips, the absdute hias for predicting 

dimekr mged fmm 0.012 inches (0,2 percent of the mean) 

with the mdel by I*;ozak and otlaeus to 0.22 inches (3.2 percent 

of the mean) with Omerad's model (Table lo), Mean absolute 

difference was about 11% inch for dinmebr witb $1 five models, 

Absoluk bias in height prediction ranged from 0.028 feet (0.1 

percent of the me=) with Denrraerselka-lk's model to 2.7 feed 

(12,5 percent of the mem) wI.L;k Omerod$ msdel; me= absolute diffe~ace ranged fmrn 3 to 5 feet, Tho eoefBefent of vadation was tibout 100 percent for dimekr and height for alf Eve 

models. With volume prediction, absolute bias ranged from 
Table 9.-Bias,' mean absolute difference (m), and standard deviation of the difference (SD) of the 

various models for six independent sets of data 

Test Dimeter outside bark 

Model site Bias sn 

Height Volume ineluding bark 

Bias r;;l SD Bias SD 

---..--..-*- feet - - . - - - - - - - -----., cubic feet- - - - - - - 

Bmce and 1 ,156 '342 321 

others 2 -087 .428 .448 

3 .I34 .383 .446 

4 .I78 ,434 .483 

5 '383 .GO5 .667 

6 .050 -375 .448 

Kozak md 1 

others 2 

3 

4 

5 

6 

"hsitive bias - overestimation; negative bias =. underestimation. 

0,003 eubie feet (0.1 percent of the mean) for the model by 

Bruce and 0thel.s do 0-19 cubic feet (4.7 percent of the mem) 

with 0meuod9s soclef, Mean absolute difference was bebeen 

0.3 and 0,37 eubie feet svitkr a cwfficient of va~ation approach 

ing 200 percent. 

In a hul;lrer attempt to deternine the best model, the aecarriley 

and p~cision of each model were exmimed separately for 

difguent portions of the bole, me comp~sons p~vious1y desedbed, using the six independent sets of data, were repealed 

th~e times. First, only the butt pieees (of any length) from the 

six test sites we= used in the cornpairscans, Second, only butt 

pieees G 42.3 feet in IenGln were used. hd , third. only upper 

pieces (of my len@-t&) wem used to compare the models. To 

keep from burdening the mader with too may nurmbrs, if have 

summadzed the results from these addition& tests in Tabfe 10, 

Weighted mean vdues for the thlhree test edte~a, computed over 

ft appeas from the results in 'f"able 9 that the effects of test 

lociation were minim&, In generrali, the models we= less 8ecumLe 

and less puc?ciise on the trees from site 5. The trees from this 

apea were larger, dthougb mean dbh md mean total Ibeigbt were 

not simibcmtly peatea: than on site 2.. 
Table ?@.--Bias,. mean absolute difference (m ), and standard deviation of the difference (SO) of the 

various models averaged aver all six test sites for different bale sections 

Test pieces Diamekr outside bark Height 't%olrrme including bark 

gfilodef usrz-d Bias SD Bias SD Bias SD 

- - . - .. - - -inchef; - - - - - - - - - - - - . - - - - -feet - . - - - . --- ------ cubic feet- - - - - - 

Bvuee md ~ll" -169 .434 .498 -807 3.086 3,644 .003 -304 .628 

0the;l.s- Butts - any ,248 -500 .540 ,604 2.099 2,690 -.045 -547 .890 

Butts < 1 2.3 ft .229 .426 '522 .791 1.841 2.378 -.I39 ,241 AOO 

Uppers - my -1 18 .391 .464 .94P 3.734 4,023 -034 .I45 265 

Deumslarj~elaalk M1 

Butts - any 

Butts G 12,3 ft 

Uppers - my 

Moz& and All 

others Butts - any 

Butts 4 12.3 ft 

Uppers - any 

Max and All 

Burkbart Butts - any 

Butts 4 12-3 ft 

Uppers - my 

Omerod All 

Butts - my 

Butts < 12.3 ft 

Uppers - any 

"Positive bia - overestimation; negative bias = underestimation. 

"~11 pieces - mem vdues from Table 9. 

all six test sites, as well as weiighkd mean values bm Table 9, 

are presented. 

p~diction (all sections of the bole) the ""best" model depends on. your objective, me model by Kozak and others 

was the best prcsdictor of dimeder, This model was the most 

accurate (Lowest bias) and most precise (lowest standard 

deviation). For p~dicting height, however, Max and BurkhM's model did the best job by all three e~teriia, whereas the 

model by Bmce and othea was better for predicting cubic 

foot volume. men the ranks were summed for diameter, 

height, and volume, the least bias (most accurate) models for 

overall prediction were Demaersehalk's, Max and Burkhart%* 

and the one by Kozillr and others. The model by Max and 

Burkhart was somewhat more precise overall. When the thme 

c~teriare considered together, the models by Bruce and 

others and Wax and Burkhart finished in first place. The 

models by Dernaerschalk trnd Kozak and others ranked 

next, followed by Omerod's. 

Looking at the values in Table 10, it is evrident that for the 

most part the models perfomed best in the upper pa& of 

the bole and poorest in the lower butt section. lxr the upper 

bole, the actud diffeuenee mong the models was vevy small. 

Except for bi~ in height prediction, the= wa no signiGearzt 

difference moag tine Eve models for any of the remrtiniag 

er;itre~a in the upper part of the bole, Tkis may indicate that 

taper is relatively uniform in the upper bole of havdwood 

trees, even though a wide vaPiety of species and sizes are considered. On the other hand, predictions for butt sections of 

the bole resulkd in the greatest difference among the 

models. 

Next, the mean values in Table JO mre ranked from 1 (lowest - best) to 5 (highest - poorest). The results of this ranking process are shown in Table 11. We see that d'or overall 

If we look at how well the various models predicted values 

far butt pieces of any lengdh, we find that the rankings lead 
Table It-Rankin* of the five models for diameter, height, and volume prediction 

All, pieces Butt pieces Butts < 12.3' Upper pieces 

Model Rmkings for Rankings for Rantkings ;for Rmkings for 

Slam Sum Sum Sum 

Bias SD Bias SD Bias SD Bias SD 

Bmce and others 

Demaeusehalk 

Kozak md other?; 

Ma and Burkltart 

Omerod 

Bmee and others 

Demae~ehdk 

Kozak and others 

Max md Burkhart 

Omerrod 

Bmee and others 

Demlae~ehalk 

Rozak md othes 

Max and Burkkart 

Omerod 

Bmee and others 

Demaerschdk 

Kszak and othem 

Ma and Burkharl, 

$)merod 

DPmETER OUTSIDE BARK 

HEIGHT 

VOLUME INCLUDING BARK 

SUM OF RANKINGS 

Its nearly identied afcsrachrrsions. About the only change was 

that the difference between the top four models was less 

pronounced. Thus, so far, there seems to be little ratiortde 

for proclaiming one model superior. 

However, if we iook just at the more extreme portions sf the 

bole, upper pieees of any length, and butt pieces G 12.3 feed, 

it is evident that Max and Burckhart" model ranked highest 

in most categories. Their rmdel was more accurate (less 

biased) for dl p~dictions in the Iglwer 12.3 feet sf the bole, 

and the precision of the model ra~ked high as well, However, even though Max and Brrrkhartk sodel ranked at the 

top for upper-bole predictions, this was mainly due to ids 

consistent perfommce; because, as noted earlier, there was 

no sipigcmt difference among the Eve models in this portion of the tree, 

Discussion md Application 

Although the study results do not establish an indisputable 

"best9' htager/voiume moddisk for Appatachim hardwoods, they 

do show some general trends, Based on the regession analyses using smple data (Table 71, the independent tests with 

data from six different stands, and our knowledge of the 

modelsa esmplexities and limitations, we cm make some 

qualified recommendatiol-ss: 

(I) If the computations will not present a problem, use 

Max and Burkkaart" model, Although this model did nod 

rank highest in Ah cases, it was the most consistent performer. And it was pwticulady good, compared to the other four 

models, for predictions in the lower bole. However, it is not a 

sirnpke model to use; therefore, a computer (and the neces- 
savy software) or at least a progrmmable cdcufator with suf- (1917) indicak that a single option is not adequak for dl 

ficient nnewro~;)~ is requiwd for efficient calculations. hardwood species. And a singe option may not be satisfae- 

(2) If, on the other hand, you can saeuiGce some aeeurmy in tory for trees of the e species from diffemnt geographicat 

the lower bole to gain simplicity in use, then the model by areas. A summq o mmendations for several species is 

Kozak and othe~s would be a good choice. Overaff, their presented below: 

model, while not too precise, was as aceurak as Max and 

lE3urfrha;rt" ((Table XI), and it is very easy to use. 

Of GOU~, a5 a user, the choice is yours; the coefficients presenkd in the Appendix tables provide considerable flexibiliw 

in desiping a taperjvolume estimating system to fit your 

needs. In selecting a model, you should consider the relative 

Wiant 

and Boehmer Colaninno 

Koeh a"d and others Rennie 

(1974) (1976) (1977) 

(STX bark option recommended) 

ma~itude of the values in Tables 9 and 10 as well as the Yellow -poplar vavious rankings. Sometimes the rankings imply greater dif- 1 1 

Red maple I - Grences than were actualIy obsewed. Hiekor~;, sp. - - 

Although the models were fitted to outside bark data, the Red oaks 

user can easily eonve& dob (dimeter outside bark) estimates White oaks 

to diameter inside bark by using one of the bark options 

from Grosenbaugh (1974). Three bark options were presenkd by Grosenbaugh (1974), in his STX 3-3-73 timber 

cmising package, for estimating diameters inside bark at any 

point on the bole using the ratio of dbb inside bark to dbh 

outside bark. The three options are b~efijr descdbed below 

(from Colarzinno and othem 1977): 

Even though a substantial amount of testing was done, there 

a= still a couple of unanswe~d questions. Sinee the independent test data contained few big trees (mem dbh was 8.1 

inches), additiond testing in the large diameters (dbh 18-t 

inches) is desirable. Butt swell in larger trees might alter the 

results, afthough at least some of the taper models should 

be more accurak than Smdian5s fornula for volume estimaOptio'crrz I Di, =: Do, (BBH,,/DBW,,) tion. Smalian" fornula nomdly ove~stimates the volume 

Bark thielmess is assumed to be a constant proportion of dob of butt logs fmm large trees. 

throughout the height of the tree. 

@~~QPZ 2 Dib - Do (f .O - (1 .O - DBHib /Df3Ho )(l-0/ This leads to the second question: How elose are taper-based 

(2.0 - Doh /DBHob 111 volume estimates to the true volume? Since all of the tests 

Implies the proportion of bark decreases hyperbolically up tineluding those in most studies) have 

the tree. one estimak ag~nst another, we really do not know Wowever, cumnt research at the Forestvy Seienees Laboratory in 

Option 3 Di, = DOb (DBHib /DBHob)(S.q Puinceton, West Virginia, using water displacement tech- 

(10.0 - Dob/DBHob )) niques, should soon provide the answer. 

Implies the proportion of bark increases hyperboliedly up 

the tree. 

DBHibJDBIZOb ratios can be determined by rneasuving bark 

thickness in the field or by using average vdues detemined 

from the data used in this study (Appendix, Table 17). ?"o 

estimate height above pound (H) to a specified dimeter 

inside bark (dib), the dib value must fimt be conve&ed to 

diarnekr outside bark using one of the bark options. 

However, if bak option 1 is satisfactow, the DBH,, values 

can simply be used whenever inside bark predictions of 

dimeter, height, or volume are dedred, mis eliminates the 

need to convert any of the find estimates. 

One problem is Ghat total height is one of the required variables. Often this is not measured when timber is cmised; but 

it should be. It is the least amb'rpous height meilsurernent on 

a tree (no guessing about the 4- or 8-inch m&, etc.); and if a 

taper-basd system is used, it is the only height measurement 

needed for most trees (height to where the bole b~aks up 

would atso be required on some trees). Actually , measurfng 

total height is a small price to pay for having a complete, aecurate, and consistent estimating system for bolewood. For 

some uss, an acceptable alkmatiue would be to measure 

total height on a subsample of the cruisd trees. These data 

could be used to construct a tot&-heightldbh came for estimating the heights of the remzlining trees. 

All three options are p~sented because, so far, no one seems Any of the taper models em easily be used to prep= a 

sure which option is mmt appropriiate for a pavGicular hard- va~ety of different volume tables: total volume or merchantwood species, Mthough Option 1. is probably adequak for able volume, inside or outside bark, for an individual species 

most hardwoods, studies by Wiant and Koch (1974), or all species combined, ete. mese are just a few of the opBoehmer and Rennie (1976), and Colaninno and others tions avglable do the user. Four exmple volume tables are 
included in this repod (Appendix Tables 18 - 21) to show 

what can be obtained just by changing the upper and lower 

limits (HU and HL) of volume edculation. Sho& computer 

prosams that will generate volume tables such as presented 

hem have been developed for each of the five models. Sou~e 

listings and input instmetions for any or at1 programs are 

available upon request from the author. Mote that volumes in 

Table 20 plus those in Table 21 equal the values in Table 19. 

This demonstrates the consistency of taper-based volume calculations. Note also that certain dbh-height combinations in 

Tables 18-2 1 are obviously unrealistic; however, these are 

pemitted to simplify the programs and avoid armments over 

arbitray cutoff points. 

As we have seen, a taper-based system, regardless of the 

model usd or the goal in mind, provides accurate and consistent estimates of diameter, height, and volume for Appalachian hardwoods, Some models perform dightly better 

than others, but at the expense of simpliciw. However, with 

the infomation provided (Tables 12 - 16), the user has considerable latitude in choosing the model and the coefficients 

that best suit his needs. 

Literature Cited 

Bennett, F. A,, B. F. Swindel. 

1972. Taper cuwes for plmkd slash pine. USBA For, Sew. 

Res. Note SE-179.4 p, 

Boehmer, W, D., and J. C. Rennie. 

1976. Predicting diameter inside bark for some hardwoods 

in west Tennessee. Wood Sci. 8(4):289-212. 

Bruce, I)., R. O. Curtis, and G. Vancoeve~ng. 

1968. Development of a system of taper and volume tables 

for red alder. For, Sci, 14(3):339-350. 

Gao, Q. V., H. E. Burkhart, and T. A. Max. 

1980. Evaluation of two methods for cubic-volume pmdiction of Loblolly pine to any merchantable limit, For. Sci. 

26(1):71-80. 

Clutter, J, %a. 

1980. Development of taper functions from variable-top 

merchantable volume equations. For. Sci, 26(1): 117-120, 

Golaniinno, A,, 6. E. Sheetz, J. 6. DeMoss, and H. V. Wiant, 

Jr, 

1917. Which STX bark option is best for certain Appalachian 

hardwoods"?outh. J, Apgl, For, 1(4): 25. 

hmaerschdk, J, P. 

1911. Taper equations can be converted to volume equations 

and point sampling factors. For, Chron. 45(6):352-354. 

Demae~chalk, J. P. 

1 97 2. Converting volume equations to compatible taper 

equations. For. Sci. 18(3): 24 1-245. 

Demaerschalk, J. P. 

1973a. Integrated systems for the estimation of tree taper 

and volume. Can. J. For, Res, 3(1): 90-94. 

Dernaerschalk, J. P, 

1973b. Compatible tree taper and vofume estimating systems. Ph.D. diss. Univ. British Columbia, Faculty Fbr., 

Vancouver. 13 1 p. 

Demaerschalk, J. P., md A. Kozak. 

1977. The whole-bole system: a conditioned dualequaGon 

system for precise pmdiction of tree profiles. Can. J. For. 

Res. 1:488-497. 

Fkies, J., and B. Matem. 

1965. On the use of multivaviate methods for the construetion of tree taper euwes. Advis. Group of For. Stat. of the 

IUFRO Section 25. Pap. No. 9, Stockholm Conf., October 

1965.32 p. 

Goulding, C. J., and J. 6. Munay, 

1976. Polynominal taper equations that are compatible with 

tree volume equations. New Zealand J. For. Sci. 5(3): 313- 

322. 

Grosenbaugh , L. R. 

1974. STX 3-3-73: tree content and value estimation using 

vavious sample desips, dendrometry methods, and V-S-L 

conversion coefEcients. USDA For. Sew. Res. Pap. SE-117. 

112 p. 

Hilt, D. E. 

1980. Taper-based system for estimating stem volumes of 

uplilnd oaks. USDA For. Sem. Res. Pap. NE458.11 p. 

Kozak, A., and J. H. 6, Smith. 

1966. Critical analysis of multivariate kchniques for estimating tree taper suggests that simpler methods are best. 

For. Ghron. 42(4):458-463. 

ICozak, A,, D. D. Munro, and J. H. 6. Smith. 

1969, Taper functions and their application in forest inventory.. For, Chron. 45(4):278-283. 

Ma, T. A,, and H. E. Burkhart. 

1916. Segmented polynominaf regression applied to taper 

equations. For. Sci, 22(3):283-289. 

Omerod, D. W. 

1953. A simple bole model. For. Chron. 49(2): 136-138. 

Wiant, H. V., Jr,, and 6. B. Koch. 

1974. Predicting diameters inside bark from outside bark 

measurements on some Appdachian hardwoods. J. For. 

72(12):175. 
Parmeters for Apgdaehian Hilvdwocads 

Taper and Volume Functions 
bhl 

QFB MAPLE 

smwa MAPLE 

SYFFI PIRCH 

VELLCJ RIRCH 

WiCYbPYr Spr 

AM, REETH 

rrt.iITE dSh 

VELLOC-POPLAG 

C(JCL!P-"PFFTREE 

SLACK TUPFLO 

SLbCK CWERPY 

kJMITF 0P.K 

SCARLET OAK 

CHESTNUT OAK 

HFD ObV 

XbCK OdK 

PLbCd LOCU5T 

4H, R4SSWP00 

PhBLE 12, PARbWEIERS FOP EqSTERN HdRDWOOO TdPER AND VOLUME FUNCTIONS, MODEL: 

DQ*2/DRH**2 = F(DRHeTHrX1 --- A POLVNOM~AL 9v BPUCFs ET AL* (19681, 

ESIIMPTED VALUES OF 0 BRF OUTSIDE R6PK. 

---- REGRESSION COEFFICIENTS 

9 2 t3 3 ~e 

0,1319 

0,1000 

Oval160 

0,1527 

0,1443 

Cieli70 

0,1274 

O*ll?b 

0,0845 

0,0839 

0,1265 

0 e 090h 

0,1167 

0,1396 

0,1092 

0*1071 

0,1116 

6,0977 

Om1 IRA 

T48LE 13. PbRAMETERS FOR EbSTERN HARDWOOD TAPER AND VOLUME FUNGTI0N5, WOOEL: 

D**2/08H**2 = 10.6**(?.0"606 8 BBH**62.O*P1-2.01 * (TH-H1"*12,0*R21 

* TY**(2,Q*BJ1 --- DEYAERSCHBLK ( f 9721, 

ESTIVATED VALUES OF p ARE OUTSIDE BARK* 

YO, OF ---- REGRFSSION COEFFICIENT< - - - - - ST ANBAR3 R 

SPECIES ORSERV, RO Fa 1 A2 R3 ERROW cQciA9E 

4LL 9796 

RED MAPLE 6083 

SUGAR MAPLE 4194 

SWEET BIRCH 512 

YELLOW HIPCH 528 

HIGYQRY* SP, 180 

&Me REECH 496 

WHITE &SF 560 

YELLOW-POPLAP h00 

CUGUMBERTqEE 448 

RLACK TUPELO 464 

PLACK CHFRRV 624 - 

HHITE CI~K 6-72 

$CARLET OAK 392 

CHESTNUT OdY 

RED OAK 576 

HLPCK OAK 360 

RLACK L3GUST 6RO 

AM, SASSWOOC sf?& 
r&@tE 14r PBRAMETERS FOR EASTERN HdROWOOD TAPER AND VOLUME FUNCTIONS, MODEL: 

O@@2fDBM*@Z z 81@(WITW-1r)+W2@1M@QC2ffW**2~1e8 -- KOZAKI Ef At (196Q) a 

ESff@AIEO V6LUES OF O ARE OUTSIDE BARK* 

NO, OF REGRESSION COEFFICIENTS STBNOARO 

SPECIES 013SEBV , B 1 82 ERROR SOtJARE 

ALL 

RE0 MAPLE 

SUGAR MAPLE 

SHEET BIRCH 

YELLOW RIRGH 

hfCKORV9 SP, 

&Me BEECH 

WHITE bSH 

VELCOb-POPCAW 

GUCUMPERTREE 

RCbCK TUPELO 

@LACK CHERRY 

WHITE OAK 

SCARLET OAK 

CHESTNUT OAK 

RED OAK 

RbACY OAK 

BLACK LOCUST 

AH. BASSMOO0 

4L.L 

RFD lJdPLE 

SUGOF; VAPLF 

SWFET 9TRCti 

VFLLOM BiQfH 

HfCKOuVr SQ, 

4M, SFFCI? 

WITE asn YELLOW-POPL a+ 

CllCCfMYERTnEE 

BLACK TUPELO 

FJLACK CHERQY 

WHITE OAK 

SC~RLET 0at.r 

CHESTNUT O~K 

Rh-r? OBIc 

HLbCK OAK 

QLACK LOCUSP 

nth, PsssrnnD 

TdRLE 15, PdHbMETERS FOR EASTERN HARDWOOD TAPE@ dND VOLUME FUNCTIONS, MODEL: 

0**2/09H*Q?=F(DRXeTH*Wl - SEGMEhTEO POLYNQMIAC, NbX 6 BURKWART (lY7h) 

ESYIMbfEr) VBLUEI; OF D ARF Ot!lS19E 9AF)Ka 
T~RLE 16, PARAMETERS FOR EASTERN HARDWOOD TAPER Ah0 VOLUME FUNCTIONS, MODEL: 

D**?lDBH*@? = ~1TH-W)/flY-4,Sf)@"~2,00B1f --- ORMEROO t 1973 1 , 

ESTIPATEO VblUES OF f3 ARE OUTSIOE RaRKe 

YO, OF RFGRESSION COEFFICIEWT 

SPECIES ORSERV, H 1 

ALL 9296 

QED M)LDLE 608 

SUGAR MAPLE 504 

SkEET RIQCY 512 

VFLLOW BIRCH "58 

HICKORY, SPe 480 

bFA, RFFGM 496 

&5'HIIF ASM 560 

VELLOW-DOPLbU hO(! 

GtlCUWPFuTMEF 4 4 9 

RLbCK TilPELO 464 

RL~CK C~EMRV 624 

M~ITE Ol?K 672 

SCARLFT 04Y 392 

CHFST~UT OAK 4RR 

PFD OAK 576 

 LACK QBK 360 

MLdCK LOCUST 480 

bM, BbSS~000 504 

STANDAgD 

ERROR 
Average Dbhib [Dbh,, Ratios 

for the Sample Data 

Exarnpfe Volume Tables Prepaved wid-h the 

Taper Model by Koz& and Others* 

@ Totd volume - Table 18 

@ Merchantable volume to $-inch top - Table 19 

@ Merehmtable volume to 8-inch top - Table 20 

@ Volume between 8- and $-inch points - Table 21 

*All volumes are inside bark. Bark option I was used to 

make the eonversion. 
Tab7eIfj. --GHC)SS PEELED VOLUME IN CUBIC FEET (EXCLUDING BARK BY TOTAL HEIGHT, 

NOUEL BY KOZAk, ET ALe (Y = D*@2/O8H@*2) USING COEFFICIENTS FOR ALL SPECIES 

VOLUM~ dETWEkN LOWEH LIMIT OF 0.0 FEET 

AND UPPER LIMIT OF 0.0 INCHES 

30 

FEET 

TOTAL HEIGHT FRO# GROUND-LINE TO TIP : 

40 50 60 7 0 80 90 100 

FEET FEET FEET FEET FEET FEET FEET 

110 120 

FEET FEET 
labSe14b --GROSS PEELED VOLUME IN CUBIC FEEI (EXCLUDING I"-IBRKI t3V TOTAL HEIGHT, 

WOOEL BY KOLAKI EI Ale (V = D**2/DBH"*2) USING COEFFICIENTS FOR ALL SPECIES 

VOLUME BEfkEtN LOWER LIMIT OF 2.0 FEEI 

AND UPPER LIMIT OF 4+0 INCHES 

TOTAL HEIGHT FROM GROUND-LINE TO TIP : 

DBHOR 30 40 50 60 70 80 90 100 110 128 

ffnts) FEET FEET FEET FEET FEET FEET FEET FEET FEET FEET 
Table 20,--GROSS PEELED VOLUME IN CUBIC FEET bEXGLUOIEJG BARK1 BY TOTAL HEIGHT, 

MODEL BY KOZAKT EI Ale fY = 0**2/08H**21 USING COEFFICIENTS FOH ALL SPECEES 

VOLUME dEIWEEN LOWER LIMIT OF 1,0 FEEI 

AND UPPER LIMIT OF 8e0 INCBES 

TOPAL HEIGHT FROM GROUND-LINE TO TI@ : 

OBr401J 30 4.0 50 60 70 80 90 100 110 120 

4 Ibis 1 FEET FEET FEET FEET FEET FEET FEET FEET FEET FEET 
Table21.--GHOSS PEELED VOLUME IN CUBIC FEET fEXCLUDIr4G SARKf BY TOTAL HEIGHT. 

VODEL BY KOZAK, ET AL. fY = D*"2/0BH**2) USING COEFFICIENTS FOH ALL SPkCIES 

VOLUME BETWEEN LOWER LIMIT OF 8.0 INCHES 

AND UPPER LIMIT OF 4.0 INCHES 

TOTAL HEIGHT FHOM GROUND-LINE TO TIP : 

C1BhOB 30 40 SO 60 70 80 90 100 110 120 

(IN* 1 FEET FEET FEET FEET FEET FEET FEET FEET FEET FEET 

'u.s. GOVERNMENT PRINTING OFFICE: 1981-703-01 1/29 
MarBn, A. Jeff. Taper and volume equations for selected Appdaekriaa haudwood species. BroomaH, PA: Nodfieast, For. Exp. Stn.; 

1981; USDA For. Sem. Res. Pap. HE-490. 22 p, 

Coefficients for five tapc?r/volume models are developed for 18 Appdaelnian hardwood species. Each model can be used to esl;irmate 

dimeter at my point on the bole, height to any p~sefeeted dimekr, and cubic foot volume bet-vveen any two points on the bole. 

me wsulting equations we= bstecl on six sets of independent data 

and an evduation of these ksts is included. A wide va;eiiety of volume tables can k emstmeted with the models; some exaples are 

ODC: 524-1 

Keywords: Taper, tree volume, hardwoods