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Forest Scœ,vol 31, No. 2, 1985, pp. 478-484
Copyright1985, by the Societyof AmericanForesters
IncorporatingCrown Ratio Into Taper Equations for LoblollyPine Trees Harold E. Burkhart and Sally Burton Walton ABSTRACT.Analyses were performed to ascertainthe feasibility of incorporatingcrown ratio, with dbh and total height,into taperequationsfor loblollypine treesfrom unthinnedplantations. Crown ratio was related to the parameterestimatesin a simple quadratictaper model, to the join points in a segmentedtaper model, and to a singletaper coefficient.The reduction in the error sumof squaresdue to addingcrownratio afterdbh andtotal heightwasslight,andit wasconcluded that, in many cases,the inclusionof crownratio as a predictorvariablewould not be warranted. FOREST SCl. 31:478-484.
ADDITIONAL KEYWORDS.Pinustaeda,volume,form, regression analysis.
TREESTEMSare used for a variety of products; consequently,considerableeffort has been devoted to developingaccuratevolume estimation proceduresfor any standard of utilization. Taper equations provide a decription of the entire stem profile, thus allowing computation of volume for any specifiedproduct and size-classdefinitions (Avery and Burkhart 1983, p. 98). Most taper equationscomputed in the past have used diameter at
The authorsare Thomas M. BrooksProfessorand Former GraduateResearchAssistant,respectively,
in the Departmentof Forestry,Virginia PolytechnicInstituteand StateUniversity,Blacksburg, VA 24061. S. B. Walton is now Inventory Analyst with WeyerhaeuserCompany in Tacoma, Washington. Manuscript received 30 December 1983.
478 / FOREST SCIENCE
breast height (dbh) and total tree height as predictor variables. There is a limit, however, to the precisionwith which models using only dbh and total height can predict tree taper becauseof the effect of other tree and stand variables. In caseswhere a high degree of precisionis required,it may be desirableto introduceadditional variablesto improve taper prediction. The objective of this study was to explore methods for incorporatingcrown
ratio (lengthof live crown divided by total tree height)into taper modelsthat have been applied successfullyin the past with the predictor variablesdbh and total height. EFFECT OF CROWN SIZE ON BOLE FORM
Larson (1963) noted that most variations in bole form are attributed to changesin the size of the live crown, its distribution along the stem, and the length of the branch-free bole. In a study of the effectsof brancheson diameter growth of loblolly pine, Labyak and Schumacher(1954) found that the number of branchletsand the location of a singlebranch on the bole determine its contribution to diameter growth of the main stem. They determined that brancheslower in the crown contribute to diameter growth almost uniformly throughout the stem. In contrast, a branch in the upper one-tenth of the tree contributes to cross-sectionalarea growth essentiallyjust below the base of the branch. Results also indicate that a branch below 50 percent of tree height with fewer than three branchlets contributesnothing to diameter growth of the main stem. Kozlowski (1971) cited a study which verified the concept that crown size affectsradial growth below the crown but has less effect on growth within the crown. Farrar (1961) studied the distribution of diameter growth along the stem for trees in several positions in a stand. He found that the thicknessof the outermost annual ring in
a healthydominant tree in a standof polewoodsizetimber with a medium densityvaried as follows:The ring was narrow at the tree tip, but increasedin width in the descentthrough the crown to a maximum near the brancheswith the most foliage. The annual ring width decreasedin the lower crown and for a major part of the bole and widened again in the butt. The annual ring width of an open-grown tree followed the same pattern as the dominant tree, but there was more growth throughout the stem. A crowded tree was characterizedby lesscross-sectional growththan the dominant, maximum ring width closer to the tip, and lessthickeningat the butt. Newnham (1965) found that taper increasedthroughoutthe life of a tree as long as it could be classifiedas dominant. Generally, when a tree became suppressed,its stembecame more cylindrical with age.Kozlowski (1971) cited severalstudieswhich demonstratedthe strong tie between crown size and the point along the stem of maximum annual ring increment. This point was low in youngtreeswith long crownsand moved upward as low branchesbecamelessvaluable. Dell (1979) discussedthe potential of usingcrown ratio to refine tree volume or product yield estimates.Despite wide recognition that crown size influencestree taper, few studieshave incorporateda measureof live crown sizeinto taper prediction. Feducciaand others(1979) and Dell and others(1979), working with loblolly and slashpines, respectively,divided their data into three crown ratio (CR) classes(CR < 36 percent; 36 percent < CR < 50 percent; CR > 50 percent) and estimated separatesets of coefficientsfor eachof the groups.The treesusedto estimatemodel parametersspanned a broad range of dbh, total height, and crown ratio combinations. Results from these separate, independent fittings showed that, holding dbh and total height constant, trees with smaller crown ratios generally had greaterpredicted volume. This result was expected becauselower crown ratios imply more cylindrical stems with more volume. Although
improvementsmay be realizedby dividing data into classesand fitting separateequations, information in the data is lost in the processand anomalousresultsmay be obtained for certain valuesof the independentvariables.It is generallypreferableto incorporatesources of variation into prediction equationsas continuousvariables.The analysesreported here were aimed at incorporatingcrown ratio, with dbh and total height, asa continuousvariable in taper prediction. DATA
Data usedin this studycame from tree measurementson loblolly pine treesfrom old-field plantationsin Virginia, North Carolina, Maryland, and Delaware. These plantationshad
VOLUME 31, NUMBER 2, 1985 / 479
TABLE 1. Distribution of sample treesby age and stand density(numberof treesper acre at time of observation)classes.
Age class (years)
Number oftrees per acre class < 501
501-800
801-1,100
1,101-1,400
> 1,400
Total
............................................................. Number of sampletrees.............................................................. 8-10
2
42
18
2
2
66
11-15
14
98
68
16
2
198
2
160
16-20
8
106
44
21-25
10
18
4
26-30
4
4
8
31-35
12
2
14
Total
50
270
134
2
20
34
6
480
not been thinned, burned, or pruned, and were free of severe insect or diseasedamage. A total of 480 trees were selected,felled, and measured on temporary yield plots. On each plot, the tenth and twentieth trees measured for dbh were designatedsample trees and felled with an approximate stump height of 0.5 feet. (Although actual stump heightswere not recorded, a constant height of 0.5 feet was assumedin these analyses.)The dbh, total height, and length of live crown were measured and the stem was cut into 4-foot sections to a top diameter of about 2 inches, outside bark. Diameters inside and outside bark were recorded for the stump and the top of each 4-foot bolt. The sample trees ranged from 3 to 12 inchesin dbh (average6.4 inches),from 20 to 80 feet in total height (average43.6
feet), and from 20 to 80 percentin crown ratio (average45.0 percent).Table 1 showsthe distribution of the sample trees by age and stand density (number of trees per acre at time of observation) classes.Additional information about the data can be found in Burkhart and others (1972). ANALYSES
The potential contribution of crown ratio to taper equationswasinitially assessed by fitting a taper model in dbh and total height after dividing the data into crown ratio classesand then examiningthe relationshipbetweenmean crownratio of the classesand the coefficients of the taper model. Studying trends of coefficientsover discretecrown ratio classesmay help specify an appropriate model for inclusion of crown ratio as a continuous variable. The data were arbitrarily divided into three groups(CR < 39 percent;39 percent < CR < 52 percent; CR > 52 percent) in order that the number of trees in each categorywould be approximately equal. The widely used and relatively simple model of Kozak and others (1969) d2/D2 = bo + b•(h/H) + b2(h/H)• where
d=
diameter (inside bark) of stem at height h dbh (outsidebark) H = total tree height D=
was fitted to the data in each crown ratio classby linear regressiontechniques.As crown ratio increasedparameterestimatesincreasedin absolutevalue, indicatingthat the parametersare related to crown ratio (Table 2). The variablesCR, (h/H)CR, and (h/H)/CR were addedwith the seconddegreepolynomialin relativeheight(h/H) and allowedfreeto enter
by a stepwiseregression procedure.All three variablescontainingcrownratio enteredthe equationafter the relative heightterms (0.10 probabilitylevel), but the increasein R 2 was marginal(the R 2 value increasedfrom 0.8779 to 0.9010 by adding CR, (h/H)CR, and (h/H)/CR after (h/H) and (h/H)2).From theseresults,it wasconcludedthat either(1) the functions of crown ratio free to enter the stepwise procedure were not appropriate for
480 / FOREST SCIENCE
TABLE 2. Estimated coefftcients for Kozak and others(1969) taper modelfor threecrown ratio (CR) classes. Coefficients •
Crown ratio dass
bo
b•
b2
0.00 < CR < 0.39
0.9955
-1.9608
1.1196
0.39 < CR < 0.52 0.52 < CR < 1.00
1.0327 1.0882
-2.0509 -2.3231
1.1410 1.3814
Model:d2/D2 = bo+ b,(h/H) + b2(h/l-l) 2,variablesdefinedin text.
incorporatingthe effect of crown ratio, or (2) the effect of crown ratio on parametersin this particular model is not marked. In a secondattempt to incorporate crown ratio into a taper model, a somewhat more complex equation form was used. Becauseattemptsto describea completetree stem with one function have usuallyresultedin poor estimatesof diameter in the regionof butt swell and stem tip, Max and Burkhart (1976) used a segmentedpolynomial regressionmodel which describesthe butt, main stem, and tip of the tree as three separatesolidsjoined at appropriate locations. Kozak and others' (1969) taper model was used for the submodels. The overall model may be written: d2/D2 = b•(h/H - 1) + be(hVI-? - 1) + ba(a• - h/H):I• + b4(a2-- h/H)212 where
bi -- submodel coefficients ai = submodel join points
Ii = 1, if h/H-< ai 0, if h/H > a• and other variables remain as previously defined. When coefficientsof the model were estimated using the three crown ratio groups,the nonlinear least squaresprocedurefailed to convergefor the first group (CR < 39 percent). Becauseno trends could be detectedwith only two groups,the data were divided into nine crown ratio groups, each containing approximately equal numbers of trees. Convergence was obtained for all groups except the groups with the smallest crown ratio (CR < 32 percent) and the largestcrown ratio (CR >- 63 percent). There were no discernabletrends
TABLE 3. Estimated submodeljoin pointsfor Max and Burkhart (1976) taper modelfor nine crown ratio (CR) classes.'
Crown ratio class
0.00 --
