Polymorphic Site Index Equations for Loblolly Pine Based on a ...

Forest Sci., Vol. 28, No. 3, 1982,pp. 544-555 Copyright1982,by the Societyof AmericanForesters

PolymorphicSite Index Equationsfor Loblolly Pine Based on a Segmented Polynomial Differential Model JAMES S. DEVAN HAROLD

E. BURKHART

ABSTRACT. Site index equationsfor old-fieldplantationsand naturalstandsofloblolly pine were producedfrom the integratedform of heightincrementmodelsfitted by segmentedregression techniques.After transformingthe stem analysisdata as the natural logarithm of height and the inverseof age, heightincrementsubmodelswere fittedfor variouscombinationsof the independent variablesheight, age, and age squaredfor two segmentswith a join point being a function of age. Comparisonsof the resultant curves indicated that the model having all three variables in both segmentswas superiorto other forms and to publishedequationsin predictingsite index of trees both from the data usedin parameterestimationand from an independentsource.FOREST Sc•. 28:544-555.

ADDITIONAL KEY WORDS. Pinus taeda, site quality, base-ageinvariant, differential equations, segmentedregression.

SITE INDEX CURVESare an important means of evaluating site quality. Several studies (for example, Bull 1931, Bailey and Clutter 1974, Beck 1971, Trousdell and others 1974) have shown that the shape of the height-age curve describing tree height growth varies with site quality. Families of site index curves which exhibit different shapesfor varying site quality are commonly termed polymorphic. When constructinga set of site index curves a base age must be arbitrarily chosen. It is desirable that the curves be base-ageinvariant, i.e., curves have the same shape for all choices of a base age (Bailey and Clutter 1974). The purpose of this paper is to present a method for developingbase-ageinvariant polymorphic site index curves and to show the results of applying this method to loblolly pine stem analysisdata from old-field plantationsand natural stands. PAsTWORK

Most papers concernedwith site index equationsin the last decade have dealt with polymorphic curves. Many authors have adapted the sigmoidgrowth model describedby Richards(1959). Carmean (1972) developedpolymorphiccurvesfor several oak speciesby stratifying stem analysis data into 10-foot (3-m) site index classesat a baseage of 50 years. By fitting Richards'growth equationseparately to data from each site index class, a set of curves was obtained such that curve

The authorsare former graduateresearchassistantand professor,respectively,in the Department of Forestry, Virginia PolytechnicInstitute and State University, Blacksburg,VA 24061. J. S. Devan is now employed as a ResearchAnalyst by International Paper Company, SouthlandsExperiment Forest, Bainbridge,GA 31717. Manuscriptreceived 23 December 1980.

544 / FOREST SCIENCE

shape was dependent on site index class. Polymorphic site index curves have also been developedby modifyingRichards' equation such that height is a function of site index and age (Beck 1971, Trousdell and others 1974, Burkhart and Tennent 1977, Graney and Burkhart 1973). Base-age invariant polymorphic site index curves can be derived by fitting heightincrement as a function of height and age. The heightincrement equation, depending on the model form, is then either solved as a differential equation or integrated to obtain a total height equation. The resultant curves are base-age invariant since site index is introduced into the constant of integration and not used for parameter estimation. Clutter and Lenhart (1968), in developingbaseage invariant curves for loblolly pine, used the following differential equation:• dY dX

- bo + b•X + b2Y

where

dYis thefiniteheight increment calculated fromadjacent boltsi andi - 1 dX as (In(H,) - ln(H,_O}/(1/A,- 1/A,_•} bo,b•,b2 are parametersto be estimatedby regressiontechniques. dY

X isatransformation oftheage atwhich •--ffisassumed toapply. It iscalculated as [1/Ai + 1/At_•]/2.

Yisatransformation ofthe height atwhich d•-isassumed toapply. Itis calculated as [In(H0 + In(Hi_0]/2. PROCEDURES

In additionto base-ageinvarianceand polymorphism,there are other desirable propertiesfor site index curves. For most species,the upper asymptoteshould be a function of site index with trees on better sitesreachinga higherultimate heightthan thoseon poor sites.The equationshouldgive a heightof zero at age zero, and a height equal to site index at the base age. The equationshouldalso expressan inflectionpoint if the data usedto estimatethe parameterswarrants it. The inflectionpoint would be the age at which heightincrementis maximum. In order to be useful and readily accepted,any method for deriving site index equationsshouldbe easy to apply and the resultseasy to interpret. Few models will meetall of thesecriteria,but the goalshouldbe to meetas manyaspossible. The approachused here to developpolymorphiccurvesthat are base-ageinvariant involved expressingheightincrementas a functionof heightand age. The fitted equationis then solvedas a differentialequationto obtaintotal height.In reviewing various site index models in the literature, some were found to have very goodcharacteristicsfor youngtree ageswhile othersfit better at older ages. With this observationin mind, segmentedpolynomial regression(Gallant and Fuller 1973,Fuller 1969)was used to developa systemof equationswhere each equationwould be used over a given domain. The generalform of the model is Y =f•(X) =f•(X)

0 • X • a• a• < X _--< a•

:

= f,•(X)

a,•-i < X _--< m

x Detailedinformationaboutthe site index modelpublishedby Clutter and Lenhart (1968)can be found in Lenhart, J. D. 1969.Yield of old-fieldloblolly pine plantationsin the Georgiapiedmont. Ph D Diss, Univ Ga. (Diss Abstr No. 70-01178.)

VOLUME 28, NUMBER3, 1982/ 545

where f•(X), i = 1..... n is a polynomial function of X over the i th segment. The equations are joined together at the points a•, a2..... an-• by imposing restrictionson the model. Fuller (1969) discussedestimatingthe parameters of f• with fixedjoin points, and Gallant and Fuller (1973)coveredestimatingthejoin pointsfrom the data. Segmentedregressiontechniqueswere applied by Max and Burkhart (1976) to developtaper equationsfor 1oblollypine. The requirementsthat the height incrementmodel have a closedform solution as a differentialequation,and be a polynomial,are met in the generalform dY dX

- bo + b•Y + g(X)

where

dYistheinstantaneous growth rate(estimated asafinite difference).

dX

b0,b• are parametersestimatedby regressiontechniques. dY

Yistheheight ofthetree atwhich •- occurs. dY

g(X)isapolynomial function oftheage atwhich •- occurs. To obtain a smooth total height curve, the height increment equation must be continuousat the join points. The total height curve was made continuousby using the constantsof integration in adjacent segmentsto ensure equal height values at the join point. In our research, only models with a singlejoin point in age were considered.The join point age that gave the best fit was found by fitting the model usinginteger age values and identifyingthe age with the minimum sum of squared error.

Theinstantaneous rateof heightgrowthdH/dAwasapproximated by a finite difference after making a logarithmic transformationof height and a reciprocal transformationof age. This growth rate was assumedto occur at a time and height equal to the arithmetic averages of the transformed variables. These transformations and assumptionsare the same as those of Clutter and Lenhart (1968). The polynomialfunctionof the ageat whichdY/dX occurswas restrictedto be at most seconddegree as higher order polynomialsgreatly increase the number of parametersand the complexity of the model. DERIVING

THE JOIN-POINT

MODEL

The onejoin-point modelwith a seconddegreepolynomialof agein each segment will be derived, noting that models with lesser degree polynomials are special casesof this more generalform. The assumptionis that dY/dX can be described as

dY _{bo +b•X +b2X •+b6Y

dX

ba '-I-b4X "l-bsX• + b6Y

forX=< L (1) for X > L

(2)

where X is the reciprocalof age, Y is the logarithmof height, and L is the nonlinearjoin point. Since L is not related to height, the parameter for Y is dY

thesame forboth segments. Thecondition that•- isequal forboth segments at L can be written

as

bo + b•L + b2L• + b•Y = ba + b4L + bsL• + b•Y. Solvingequation(3) for bs gives

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(3)

ba = b0 + (bl - b4)L + (b2 - bs)L2.

(4)

Substitutingequation (4) into equation (2) reduces the number of parameters in the system by one and gives

dY _{bo +b•X +b2X •+boy

dX

forX-< L(1)

bo+ (b• - b4)L+ (b2- bs)L• + b4Xq-b,X• + boy for X > L. (5)

Now considerthe following dummy variable: I = 0 if I = 1 if

X_-< L X> L

(note: 1 -I = 1) (note: 1 -I=0).

The two segmentsgiven by equations (1) and (5) can now be combined into a singlecontinuousequation using dummy variable I: dY

dX - bo+ b•X(1 - 1)+ b2X•(1 - 1)+ (b•- b4)LI+ (b•- b•)L•I + b4XI + b,X2I + boy

(6)

after collectingterms, equation(6) reducesto dY

dX - bo+ b•[X+ I(L < X)]+ b2[X • + I(L• - X2)]+ b4I(X- L) + b,I(X • - L •) + boY.

(7)

Equation (7) is the linear regressionmodel from which all six parametersare estimatedsimultaneously.The seventhparameter,ba, is then obtainedfrom equation (4) after all other parametershave been estimated?The significanceof the additionalterms due to the join point can be found with an F test on the sum of squares due to the regression.

The final total height curve is found by individually solving the equation for each segmentas a separabledifferentialequationwith an integratingfactor (Manougianand Northcutt 1973). The model with the X-squared term will be solved noting that simpler modelscan be treated as specialcases.Equation (1) can be rewritten to separatethe variablesas dY

---

dX

boY = bo + b•X + b2X•.

(8)

Theintegrating factoris e•{x•wherep(x) = J'-bodX = -boX.Thesolution to equation(8) becomes

Ye-b6x= J'e-b6X(bo + b•X + b2X•) dX + K, and

bo+ b•-+•'-•-! b• 2b2'• _(•-• b•+•-•2! 2b•'•X - b•x• •6x(9) Y=- (•-• bo +Ke where K is the constantof integration. 2 One reviewer suggestedhaving the height increment equation both smooth and continuousby imposingthe additional constraint b4 = b• + 2(b2 - bs)L.

The resulting analogueto equation (7) would be dY

•- = bo+ b•X+ b2[X • - (L - X?]+ bs(L- X?+ b6Y. Limited testingof this model showedit to be quite similar in predictive ability and fit to equation (7); with some data sets it may prove superior.There was, however, a tendencyfor this model to approach infinity as age went to zero sooner than with equation (7).

VOLUME 28, NUMBER 3, 1982 / 547

After integration, both segmentswill have the same slope but different values at the join point. The integrating constant K for the segmentthat contains the base age is used to cause height to equal site index where the age is equal to the base age. The integrating constant for the remaining segment is used to cause both segmentsto have equal values at the join point. Equation (9) is solved for K, replacingY and X with Y0and X0, giving

_O6xo/bo b•

262•

K = Yoe -•6xo + e [•-•+ b•-+ b•/

+ Xøe-•xø (b•+ 262'• b•/ + •b• b2 ,• -•6Xo A0e

(10)

where X0 is the inverse of some index age •d Y0 is the natural logarithm of some hieght at the index age. ReplacingK in equation (10) with equation (11) and simplifyinggives

Y='0z+

+

+

(Z-l)+

+

(XoZ - X)

+ b6 ,-•-(X•Z - X•)

(11)

where Z = e o6•x-xo.The analoguefor the secondsegmentwould be

r=r0z+

+b•(•Z- X•).

(Z-l)+

(x0z-x) (12)

For the case where a site index is desiredfor a given height and age, the natural logarithm of height and the inverse of age replace Y and X, respectively, in equation (11) for the segment on which the given age falls. If this segment also contains the base age of interest, the inverse of the base age replaces X0 and equation(11) is solvedfor Y0which would be the natural logarithmof site index. If the segmentdoes not containthe base age, X0 is replacedby the inverse of the join-point age and equation (11) is solved for Y0 which is the natural log of the heightat the join-point. The inverseof the join-point age and the natural logarithm of the height calculatedat the join point are used in equation(11) for the adjacent segmentto find the natural logarithm of site index (Y0). When a height is desired for a given site index and age, the inverse of the base age and the natural logarithmof site index replace X0 and Y0in equation (11) for the segmenton which the base age falls. If this segmentalso contains the age of interest, the inverse of age replacesX and the equationis solvedfor the logarithm of height (Y). If the age of interest is not on this segment,the inverse of the joinpoint age is used for X and the equation is solved for the natural logarithm of

heightat the join-point (y). The inverseof the join-point age and the logarithm of the associatedheight replaceX0 and Y0in the equationfor the adjacentequation. With X replaced by the age of interest, this equation is solved for Y which is the logarithm of height at the age of interest. THE

DATA

The data for this study were obtainedfrom 365 tenth-acre (0.04-ha) sampleplots from natural and planted standsin Virginia, North Carolina, Maryland, and Delaware. (For a more completedescriptionof the plot data see Burkhart and others 1977a, b.) On each plot, two trees with no evident damagewere felled for mea-

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0

I0

20 AGE

30

40

50

{YEARS)

FIGURE 1. Siteindex cu•es(baseage50)

•rloblolly

pinein natur• stands.

surementof total height, age, and crown class. The trees were cut into 4-foot (1.22-m) bolts and ring countswere madeat the top of eachbolt. Plantationages rangedfrom 9 to 36 years, and there were from 300 to more than 1,600trees per acre (740 to 3,952 treesper ha). The naturalstandshad an age rangeof 13 to 77 yearsand densitiesrangingfrom 100to 900 trees per acre (250 to 2,200 trees per ha). The numberof trees that were dominantor codominantand qualifiedas site index trees were categorized as follows: Natural stands

Old-field plantations

Coastal Plain Piedmont

182 18

187 107

Total

200

294

From the stem analysisdata, a seriesof age-heightpairs was derivedfor each tree. The assumptionis made, that on the average,a bolt end will fall at the midpointof a terminalleader. This bias was correctedby addingone-halfof the leader lengthfor the appropriateyear to the heightat the top of eachbolt (Carmean 1972).The leader length was estimatedfrom the averageannualgrowth of two bolts above and two bolts below the bolt in question. The data were groupedinto natural standsand old-field plantations.To deter-

VOLUME 28, NUMBER 3, 1982 / 549

c)

c)

550 / FOREST SCIENCE

mine whetherfurther divisioninto coastalplain and piedmontregionswas needed, the site index model used by Clutter and Lenhart (1968) was fitted to each of the four groups. Before integrating, this model is linear and standardF tests were used to test for differencesbetweengroups.The resultsof the F tests indicated no differencebetween natural standson coastalplain and piedmont sites. Differences between coastal plain and piedmont plantation sites were significant,but when visually compared, the two curves appeared to be similar. The final classification of the data resulted in four groups: (1) natural stands, (2) piedmont plantations, (3) coastal plain plantations, and (4) combined plantations (Figs. 1-4). EVALUATION

AND RESULTS

In addition to combinationsof join points and X-squared terms, models were fitted both with and without an intercept for a total of 12 models.The best model was selectedby comparingthe fit, predictive ability, and logical characteristics of each model for each data set.

Logical characteristicswere checkedby graphingeach equation. Fit was tested by using F statistics to test the significanceof a join-point and of including X-squared terms. Predictive ability was tested with a random sampleof 20 trees from each data group that were older than the average age for each group. For each tree, site index was predicted for each height-agepair and comparedto the actualheightat a baseage equalto the averagegroupage. An averagedifference was calculated for each equation in each data set such that

• = Y•(Spi where n is the number of height-agepairs in a data set, and Spt and SAiare the predicted and actual site index for height-agepair i. Graphs of the fitted equationsshowedthat the inclusionof a join point and fitting of an intercept in the submodelsresulted in more logical behavior than single equations or join-point equationswithout fitted intercepts. As age approacheszero the mathematicallimit of height is infinity for the model applied here? The inclusionof a join point and fitted intercept resulted in height curves that did not exhibit this upturn at as early an age as those models without join points or without fitted intercepts. The F tests showed that in all cases, introducinga join point resulted in a significantlybetter fit (pt. > F = 0.0001). Introducing the X-squared terms into the join-point model also showed a similar significantimprovementin the model. Evaluationswith the 20 trees from each data group showed that the join-point models with fitted intercepts and with both X- and Y-squaredterms in the two segmentsperformed best. A data set available from the Virginia Division of Forestry which consistedof 27 treesqualifyingfor siteindex measurements from old-fieldpiedmontplantation sites was used to compare the best join-point model with other curves.4 Using plots of anamorphic curves in Burkhart and others (1972b), unpublishedpolymorphiccurvesfrom Burkhart (1973),5and the plantationjoin-point curves(coasta This undesirable characteristic

was not considered to be a serious drawback because site index

is generallynot estimatedfrom loblolly pine tree measurements prior to age 10. Curve shapeabove age 10 did not appearto be affectedby the increasingheightas age approachedzero. 4 Dierauf, T. A., J. W. Garner, and H. L. Olinger. Resultsof 1958thinningstudy after 15 years. Virginia Division of Forestry, unpublishedreport. 5 Burkhart, H. E. 1973.Site index curvesfrom stem-analysis data. Supplementto VPI-Industry CooperativeYield Study Report #6. Unpublishedmemo.

VOLUME 28, NUMBER 3, 1982 / 551

II

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al plain and piedmont combined), site index was estimated for each remeasurement on each tree. Table 1 shows how well each equation described tree height growth over a 10-year period on sites ranging from 15 to 22 meters (base age 25 years). The join-point model underestimated the site index by 0.27 meter over the span compared to underestimates of 3.37 and 4.01 meters for the curves in

Burkhart and others(1972b)and thosein Burkhart (1973), respectively. The parameter estimatesfor natural stands, coastalplain plantations, piedmont plantations,and the combinedplantation data are in Table 2. To make the joinpoint models easier to use, they have been reparameterizedin Table 3. As an example of equationusage, the site index (base age 50) of a 17-year-old naturally growing tree, 6.4 meters tall, will be found. From Table 3 the equations would

be

Y = Yog - 13.89961(Z- 1) - 39.50272(XoZ-X) - 882.95350(X•0Z - X 2)

(1)

for X _ 1/23

whereZ = eø.27aø6(x Xo•andX = 1/age,Y = In(height),X0 = 1/indexage,and Y0= In(heightat index age). Equation(2), which is for ages less than the join point, is usedto find the join-pointheightby solvingfor Y where X = 1/23, X = 1/17,

Y = ln(6.4)

and

Z = e 9'27306(1117-1123) = 1.152917

giving Y0 = In(9.16309). Equation (1) is then used to find the height at the base age of 50 by solving for Y0 where X0 = 1/50, X = 1/23, Y -- ln(9.16309), and Z = e s'•7aø•(ma-u5ø)= 1.24323

giving Y0 = ln(18.54857).

TABLE 2.

Coefficientsto the differentialform of the one-join-pointmodel.a Plantations

Natural

Parameter bo b• b2 ba b4 b5 b5

Old-field

Old-field

Old-field

stands

combined

piedmont

coastalplain

-89.3891 1399.5960 - 8187.6810 - 54.9848 270.9469 -428.6074 9.2731

- 56.5527 295.8661 - 526.6052 - 34.4344 98.7590 -94.0382 8.9618

- 59.8311 318.8499 - 537.9385 - 25.2918 41.9917 - 17.1312 9.2851

- 54.6086 281.2840 - 506.5065 - 39.5277 125.1938 - 123.4402 8.7267

jpb

23

5

Model: dY/dX = bo + b•X + b2X2 + b6Y = ba + b4x q- b,X 2 + b6Y

5

4

X•L X•L.

JP = join point.

VOLUME 28, NUMBER 3, 1982 / 553

TABLE 3.

Coefficientsfor the integratedform of the one-join-pointmodel. Plantations

Natural

Parameter

stands

Old-field

combined

Old -field

coastalplain

a

9.27306

b

- 13.89961

c d e

19.90047

21.86071

- 882.95350

- 58.76122

- 57.93590

- 58.04516

- 3.85363

- 2.87395

- 2.27965

- 3.25732

19.24990 -46.22071

8.67824 - 10.49234

f g jpb

-39.50272

23

8.96178

Old-field

piedmont

-4.08984

5

9.28506 -4.08941

4.12508 - 1.84503

8.72671 -4.08870

18.93104

11.10482 - 14.14613

5

4

aModel:

Y= YoZ + b(Z - 1) + c(XoZ-X) + d(Xo2Z - X 2) X_-< L = YoZ+e(Z1) +f(XoZ-X) +g(Xo2Z-X 2) X>L where Z = exp[a(X -X0)] and X = 1/age,X0 = 1/(someindex age), Y = natural log of height (in meters), Y0 = natural log of someheight at the index age, L = 1/JP. b JP = join point.

The predicted site index is 18.5 meters at age 50 years. Height at a given age and site index is found in a similar manner by working back from the base age to the age of interest. CONCLUSIONS AND RECOMMENDATIONS

When comparedto other models developedin this study and with several published models, the one-join-point models for old-field plantation sites with X-squared terms in both segmentswere superior in predicting site index of both trees from which the equations were developed and trees from independent sources.

The model developed for natural stands with one join point and an X-squared term in both segmentsproved superior when predicting site index of trees in the

data sets, but no independentdata were availablefor further testing. Some of the advantagesof the models that were developed include polymorphismand base-ageinvariance.Also, at the baseage, the predictedheightequals the site index. The final equation can be solved for both height and site index, and the parametersare estimatedusinglinear regressiontechniques.This method is versatile, gives good results, and is easy to apply. One obvious disadvantage is that the model tends to produce an unrealistic "tail" at young ages, restricting its use to ages greater than 10 years.6 However, site index cannotbe reliably estimatedwith heightmeasurementsfrom very young trees. Thus this restriction should not unduly limit the utility of the model and resulting equationspresented in this paper.

6 Althoughit appearsunusualthat thejoin point for the plantationequationsis at an agethat shows illogicalbehavior, modelswithout the join point show even worse behavior at young ages. When the young age observationswere dropped from the data, the predictive ability of all modelswas significantly reduced.Perhapsthe methodof data collection(sectioningat 4-foot intervals)createda large amount of random error at young ages and contributed to this unusual behavior.

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LITERATURE

CITED

BAILEY, R. L., and J. L. CLUTTER. 1974. Base-ageinvariant polymorphic site curves. Forest Sci 20:155-159.

BEcK,D.E. 1971. Height-growthpatternsandsiteindexof whitepinein the southernAppalachians. Forest Sci 17:252-260.

BULL, H. 1931. The use of polymorphiccurvesin determiningsite quality in youngred pine plantations. J Agric Res 43:1-28. BUmCI•ART,H. E., and R. B. TENNENT. 1977. Site index equationsfor radiata pine in New Zealand. N Z J For Sci 7:408-416.

BURKI•ART,H. E., R. C. PARKER,and R. G. ODERWALD. 1972a. Yields for natural standsof 1oblolly pine. Va Polytech Inst and State Univ, Sch For and Wildl Resour, FWS-2-72, 63 p. BURKI•ART, H. E., R. C. PARKER, M. R. STRUB, and R. G. ODERWALD. 1972b. Yields of old-field

1oblollypine plantations.Va PolytechInst and StateUniv, Sch For and Wildl Resour,FWS-372, 51 p.

CARMEAN,W. H. 1972. Site index curvesfor uplandoaks in the central states.Forest Sci 18:109120.

CLUTTER,J. L:, and J. D. LENHART. 1968. Site index curvesfor old-field1oblollypine plantations in the Georgiapiedmont.Ga Forest Res CouncRep 22, Ser 1, 4 p. FULLER,W. A. 1969. Grafted polynomialsas approximatingfunctions. Aust J Agric Econ 13:35-46. GALLANT,A. R., and W. A. FULLER. 1973. Fitting segmentedpolynomialregressionmodelswhose join points have to be estimated.J Am Stat Assoc 68:144-147. GRANEY,D. L., and H. E. BURKHART.1973. Polymorphicsite index curvesfor shortleafpine in the Ouachita Mountains. USDA Forest Serv Res Pap SO-85, 12 p. South Forest Exp Stn, New Orleans, La.

MANOUGIAN,M. $., and R. A. NORTHCUTT.1973. Ordinary differential equations•an introduction. Charles E. Merrill Publ Co, Columbus, Ohio. 364 p. MAX, T. A., and H. E. BURKHART.1976. Segmentedpolynomialregressionappliedto taper equations. Forest Sci 22:283-289.

RICHARDS,F. J. 1959. A flexiblegrowth functionfor empiricaluse. J Exp Dot 10(29):290-300. TROUSDELL,K. B., D. E. BECK,and F. T. LLOYD. 1974. Site index for 1oblollypine in the Atlantic CoastalPlain of the Carolinasand Virginia. USDA Forest Serv Res Pap SE-115, 11 p. Southeast Forest Exp Stn, Asheville, N C.

VOLUME 28, NUMBER 3, 1982 / 555