Equations for predicting tree height as a function of diameter
outside bark at breast height are presented for 16 tree species
from the central western Willamette Valley of Oregon. Foresters
can use these "height-diameter" equations to avoid the time consuming
task of measuring heights of all individual trees in an
inventory, a stand exam, or a timber cruise. Equation coefficients
were estimated with weighted nonlinear regression techniques.
Because site index can influence height, alternative equations including
transformations of site index as an independent variable
also are presented for 6 of these 16 species

Hann, David W.

Wang, Chao-huan

Undetermined

ScholarsArchive@OSU

Research Paper 51SDUnbound issueD 1 l l ? Does not circulateJanuary 1988Height-Diameter Equations forSixteen Tree Species in the CentralWestern Willamette Valley ofOregonChao-Huan WangDavid W. HannFOREIf REIEARCH LASCollege of Forestry Oregon State UniversityThe Forest Research Laboratory of Oregon State University wasestablished by the Oregon Legislature to conduct research leadingto expanded forest yields, increased use of forest products, andaccelerated economic development of the State. Its scientistsconduct this research in laboratories and forests administered bythe University and cooperating agencies and industries throughoutOregon. Research results are made available to potential usersthrough the University's educational programs and through Labo-ratory publications such as this, which are directed as appropriateto forest landowners and managers, manufacturers and users offorest products, leaders of government and industry, the scientif-ic community, and the general public.As a research paper, this publication is one of a series that de-scribes a completed study or experiment or lists publications ona specific basis.The AuthorsChao-Huan Wang is a graduate research assistant and David W.Haas is an associate professor of forest biometry. Department ofForest Management, Oregon State University, Corvallis.DisclaimerThe mention of trade names or commercial products in this publi-cation does not constitute endorsement or recommendation foruse.To Order CopiesCopies of this and other Forest Research Laboratory publicationsare available from:Forestry Business OfficeCollege of ForestryOregon State UniversityCorvallis, Oregon 97331Please include author(s), title, and publication number if known.Height-Diameter Equations for SixteenTree Species in the Central WesternWillamette Valley of OregonChao-Huan WangDavid W. HanniContents1 Abstract1 Introduction2 Data Description2 Data Analysis4 Results and Discussion6 Literature Cited11AbstractEquations for predicting tree height as a function of diameteroutside bark at breast height are presented for 16 tree speciesfrom the central western Willamette Valley of Oregon. Foresterscan use these "height-diameter" equations to avoid the time-consuming task of measuring heights of all individual trees in aninventory, a stand exam, or a timber cruise. Equation coefficientswere estimated with weighted nonlinear regression techniques.Because site index can influence height, alternative equations in-cluding transformations of site index as an independent variablealso are presented for 6 of these 16 species.IntroductionIndividual tree height is useful in assessing tree volume (Wal-ters et al. 1985. Walters and Hann 1986), tree position within astand (Ritchie and Hann 1986), and stand productivity throughsite index (Harm and Scrivani 1987), but it is time consuming tomeasure accurately. Instead, foresters now can use "height-diameter" equations to predict, rather than individually measure,tree heights. Traditionally, these equations have Included onlydiameter as an independent variable (Curtis 1967). However.other stand-level variables, such as basal area and site index,may improve predictive capability (Larsen and Hann 1987). For-esters also can use these equations to indirectly estimate heightgrowth by applying them to a sequence of diameters either meas-ured directly in a continuous inventory or predicted indirectly bya diameter growth equation. The latter approach can be valuablefor modeling growth and yield of trees and stands.The objective of this study was to develop equations for pre-dieting tree height as a function of diameter outside bark atbreast height (DOB) by itself or in conjunction with site index forthe tree species found in the central western foothills of the Wil-lamette Valley:Douglas-firGrand firPacific yewPonderosa pineWestern hemlockWestern redcedarBigleaf maplePseudotsuga menziesii (Mirb.) FrancoAbies grandis (D. Don) Lindl.Taxus brevifolia Nutt.Pinus ponderosa Laws.Tsuga heterophylla (Raf.) Sarg.Thuja plicata D. DonAcer macrophyllum Pursh1Oregon white oakPacific dogwoodBitter cherryPacific madroneRed alderOregon ashWillowBlack hawthornOregon crab appleOther hardwoodsQuercus garryana Dougl.Cornus nuttallii AudubonPrunes emarginata EatonArbutus menziesii PurshAlnus rubra Bong.Fraxinus latifolia Benth.Salix spp.Crataegus douglasii Lindl.Malus fusca (Raf.) Schneid.Data DescriptionThe data for this analysis were part of the data base collectedfor developing a stand growth model for Oregon State Universi-ty's McDonald and Dunn Research Forests located in the centralwestern foothills of the Willamette Valley. All 136 stands selec-ted had not been thinned or fertilized during the previous 5 years.Within each stand, a grid of 20 basal area factor (BAF) variable-radius plots was established so that the attributes of interest ofall trees > 8 Inches in DOB could be measured. Within each vari-able-radius plot, two fixed-area subplots were established so thatthe attributes of all trees < 8 inches in DOB could be measured.Trees with damaged tops or those shorter than breast heightwere excluded from the analysis. Tree DOBs were measured tothe nearest 0.1 inch, tree heights were calculated to 0.1 foot,and site index of each stand was computed with King's (1966)site-index equations (Table 1). Because sample sizes for westernredcedar, Oregon crab apple, and western hemlock are small, theuse of predictor equations for these species must be approachedwith caution.Data AnalysisMany of the height-diameter equations presented in the lit-erature use a log-linear model form (Curtis 1967, Wykoff et al.1982). However, other researchers have found that the residualsof these log-linear equations are not normally distributed (Larsenand Hann 1987), which makes correcting for log bias difficult2TABLE 1.DESCRIPTIVE STATISTICS FOR THE HEIGHT-DIAMETER MODELING DATA SET.'Number DOB (In.) Ht (ft) Si (ft)Species of trees Mean (SE) Range Mean (SE) Range Mean (SE) RangeDouglas-fir 12777 19.6 (0.13) 0.1-96.6 95.8 (0.41) 4.6-281.4 115.5 (0.10) 80.0-147.0Grand fir 2218 10.6 (0.18) 0.1-48.3 60.8 (0.86) 4.6-216.9 115.9 (0.24) 83.0-137.0Pacific yew 153 11.0 (0.53) 0.1-31.0 32.7 (1.01) 5.1- 63.6 117.6 (0.88) 83.0-136.0Ponderosa pine 61 12.3 (0.53) 0.9-19.9 56.9 (1.90) 8.9- 83.6 112.1 (1.44) 94.0-147.0Western hemlock 48 9.1 (1.22) 0.1-33.0 57.4 (4.36) 4.6-121.1 126.1 (0.80) 107.0-129.0Western redcedar 12 17.1 (5.69) 0.1-61.6 55.7 (12.48) 4.6-120.6 120.7 (1.55) 108.0-129.0Bigleaf maple 4014 10.2 (0.12) 0.1-52.1 53.8 (0.38) 4.6-139.1 115.3 (0.19) 83.0-147.0Oregon white oak 1528 12.3 (0.21) 0.1-47.8 51.2 (0.53) 4.8-115.0 111.6 (0.32) 80.0-147.0Pacific dogwood 803 2.5 (0.08) 0.1-18.1 22.2 (0.37) 4.6- 56.4 116.6 (0.33) 84.0-136.0Bitter cherry 374 4.5 (0.12) 0.1-10.5 45.6 (1.07) 5.0- 94.4 119.6 (0.60) 83.0-136.0Pacific madrone 335 8.6 (0.36) 0.1-45.3 43.4 (1.23) 4.6-118.4 112.6 (0.66) 80.0-147.0Red alder 329 10.8 (0.35) 0.1-30.4 60.1 (1.15) 5.0=124.9 113.7 (0.65) 83.0-136.0Oregon ash 245 7.9 (0.50) 0.1-42.8 45.6 (1.68) 4.9-121.0 99.3 (0.96) 83.0-137.0Willow, 107 5.2 (0.33) 0.1-17.1 31.6 (1.33) 4.6- 80.4 113.9 (1.00) 90.0-134.0Black hawthorn 72 2.4 (0.28) 0.1- 9.1 19.5 (1.46) 4.6- 47.8 106.6 (1.71) 80.0-133.0Oregon crab apple 34 3.8 (0.76) 0.2-20.8 18.0 (1.25) 5.5- 35.6 101.0 (2.02) 86.0-134.0Other hardwoods 122 3.4 (0.39) 0.1-32.9 25.1 (1.68) 4.6-102.0 114.1 (1.09) 86.0-135.0CNDOB, diameter outside bark at breast height; Ht. total tree height: SI. King's (1966) site Index; SE. standard error.(Flewelling and Pienaar 1981). Therefore, the following nonlinearequation form was chosen for this study:Ht = 4.5 + exp(b0 + blDOBb2) [1]where:Ht = total tree height, feetbi = parameters to be estimated, i = 0, 1. 2Larsen and Hann (1987) determined that the following moregeneral equation form, which includes transformations of site in-dex, explained more of the variation for some species:Ht = 4.5 + exp(b0 + blDOBb2 + b3X) [2]where X = additional independent variables that are transforma-tions of site index. They also determined that the variances ofresiduals for Equations [1] and [2] were not homogeneous. There-fore, weighted regression was used to homogenize the variance.Comparing five different weights with Furnival's (1961) index offit, Larsen and Ham (1987) found that a weight of 1.0/DOB wasbest.For this study, Equations [1] and [2] were fit to each species-specific data set with weighted nonlinear regression techniques(weight = 1.0/DOB). Regression coefficients for both equationswere checked with a t-test to see if they were significantly dif-ferent (p = 0.05) from zero.Results and DiscussionTables 2 and 3 contain regression coefficients, weighted meansquare errors, and weighted adjusted coefficients of determina-tion for Equations [1] and [2] respectively.For Equation [1], the coefficients were significantly differentfrom zero for all species except ponderosa pine and Pacific yew.For both of these species, the slope coefficient of DOB, bl, andthe power coefficient of DOB, b2, were so highly correlated thatsignificant estimators of them could not be computed. Therefore,the estimate of b2 was rounded to the nearest tenth and then4TABLE 2.REGRESSION COEFFICIENTS AND ASSOCIATED WEIGHTED STATISTICS FOR THE MOD-ELING DATA SET, EQUATION (1].Regression coefficientsMeansquareAdjustedcoefficientof deter-Species b0 bl b2 error minationDouglas-fir 7.04524 -5.16836 -0.253869 13.9608 0.5308Grand fir 7.42808 -5.80832 -0.240317 17.4615 0.6995Pacific yew 9.30172 -7.50951 -0.100000 3.6000 0.2078Ponderosa pine 6.10183 -4.48536 -0.300000 5.6573 0.3052Western hemlock 4.80198 -2.78628 -0.692389 14.9939 0.7096Western redcedar 6.18313 -4.41172 -0.283394 2.9967 0.9231Bigleaf maple 5.21462 -2.70252 -0.354756 21.0577 0.2474Oregon white oak 4.69891 -3.39164 -0.615259 11.5461 0.3383Pacific dogwood 4.49727 -2.07667 -0.388650 12.6538 0.3548Bitter cherry 7.91843 -5.45075 -0.181704 22.9442 0.5043Pacific madrone 5.84487 -3.84795 -0.289213 14.8787 0.4054Red alder 4.77965 -2.51761 -0.559775 12.2633 0.3560Oregon ash 5.29159 -2.87443 -0.364288 14.0244 0.5896Willow 4.88361 -2.47605 -0.309050 15.7936 0.1920Black hawthorn 4.23548 -1.85815 -0.459519 9.6226 0.6410Oregon crab apple 3.07101 -0.84120 -0.989203 8.9621 0.2689Other hardwoods 5.98032 -3.70131 -0.249030 14.8670 0.5958TABLE 3.REGRESSION COEFFICIENTS AND ASSOCIATED WEIGHTED STATISTICS FOR THE MOD-ELING DATA SET, EQUATION [2].'Regression coefficientsMeansquareAdjustedcoefficientof deter-Species b0 bl b2 b3 X1 error urinationDouglas-fir 6.19343 -5.34899 -0.239085 0.221197 In(SI) 13.7456 0.5380Grand fir 7.09574 -5.83136 -0.237430 0.315883E-2 SI 17.0813 0.7061Ponderosa pine 5.66202 -4.51380 -0.300000 0.403337E-2 SI 5.2433 0.3561Bigleaf maple 4.96819 -2.72538 -0.349626 0.234155E-2 SI 20.8609 0.2545Bitter cherry 5.55308 -4.24198 -0.235550 0.991261E-2 SI 19.2375 0.5844Willow 2.77188 -1.90051 -0.401561 0.139941E-1 SI 13.0206 0.3339' X1, additional independent variable; SI, King's (1966) site index; ln(SI). natural logarithmof site Index.5fixed for each species, and b0 and bl were refit; the resultingestimates were significantly different from zero.Equation [2], which accounts for site index, provided a betterfit for 6 of the 16 species. Of these 6, the only species with anonsignificant coefficient (b2) was ponderosa pine. That coeffi-cient was fixed to the same value (-0.300000) as for Equation [1].and the remaining parameters were re-estimated.In general, we recommend use of the equation with the great-est number of independent variables for the species. However, apotential user should consider, first, the expense or difficulty as-sociated with collecting the site-index information needed to useEquation [2] and, then, whether the amount of additional varia-tion explained by the more complicated equation is worth thecost or effort.In sum, the equations developed In this study show that treeheight is strongly correlated to diameter for the 16 tree speciesin this locale. In some cases, including site index in the equationsimproved the precision of predicting tree height. These height-diameter equations provide new and useful Information about treespecies growing in the central western foothills of the WillametteValley of Oregon.Literature CitedCURTIS, R.O. 1967. Height-diameter and height-diameter-age equations for second-growth Douglas-fir. Forest Science13:365-375.FLEWELLING, J.W., and L.V. PIENAAR. 1981. Multiplicative re-gression with lognormal errors. Forest Science 27:281-289.FURNIVAL, G.M. 1961. An index for comparing equations used inconstructing volume equations. Forest Science 7:337-341.HANN, D.W., and J. SCRIVANI. 1987. Dominant-height-growthand site-index equations for Douglas-fir and ponderosa pine insouthwest Oregon. Forest Research Laboratory, Oregon StateUniversity, Corvallis. Research Bulletin 59. 13 p.6KING, J.E. 1966. Site index curves for Douglas-fir in the PacificNorthwest. Weyerhaeuser Co., Centralia, Washington. Weyer-haeuser Forestry Paper 8. 49 p.LARSEN, D.R., and D.W. HANN. 1987. Height-diameter equa-tions for seventeen tree species in southwest Oregon. Forest Re-search Laboratory, Oregon State University, Corvallis. ResearchBulletin 49. 16 p.RITCHIE, M.W., and D.W. HANN. 1986. Development of a treeheight growth model for Douglas-fir. Forest Ecology and Man-agement 15:135-145.WALTERS, D.K., and D.W. HANN. 1986. Predicting merchantablevolume in cubic feet to a variable top and in Scribner board feetto a 6-inch top for six major conifers of southwest Oregon. For-est Research Laboratory, Oregon State University. Corvallis. Re-search Bulletin 52. 107 p.WALTERS, D.K., D.W. HANN, and M.A. CLYDE. 1985. Equations-and tables predicting gross total stem volume in cubic feet for sixmajor conifers of southwest Oregon. Forest Research Laboratory,Oregon State University, Corvallis. Research Bulletin 50. 36 p.WYKOFF, W.R.. C.L. CROOKSTON, and A.R. STAGE. 1982.User's guide to the Stand Prognosis Model. USDA Forest Service,Intermountain Forest and Range Experiment Station, Ogden,Utah. General Technical Report INT-133. 122 p.British/Metric Conversion1 inch = 2.54 centimeters1 foot = 0.3048 meters7WANG, C.-H., and D.W. HANN. 1988. HEIGHT-DIAME-TER EQUATIONS FOR SIXTEEN TREE SPECIES IN THECENTRAL WESTERN WILLAMETTE VALLEY OF ORE-GON. Forest Research Laboratory, Oregon State Uni-versity, Corvallis. Research Paper 51. 7 p.Equations for predicting tree height as a function of diameter outside barkat breast height are presented for 16 tree species from the central westernWillamette Valley of Oregon. Foresters can use these "height-diameter"equations to avoid the time-consuming task of measuring heights of all indi-vidual trees in an inventory, a stand exam. or a timber cruise. Equation co-efficients were estimated with weighted nonlinear regression techniques.Because site index can Influence height, alternative equations includingtransformations of site index as an Independent variable also are presentedfor 6 of these 16 species.WANG, C.-H., and D.W. HANN. 1988. HEIGHT-DIAME-TER EQUATIONS FOR SIXTEEN TREE SPECIES IN THECENTRAL WESTERN WILLAMETTE VALLEY OF ORE-GON. Forest Research Laboratory, Oregon State Uni-versity, Corvallis. Research Paper 51. 7 p.Equations for predicting tree height as a function of diameter outside barkat breast height are presented for 16 tree species from the central westernWillamette Valley of Oregon. Foresters can use these "height-diameter"equations to avoid the time-consuming task of measuring heights of all indi-vidual trees in an inventory, a stand exam, or a timber cruise. Equation co-efficients were estimated with weighted nonlinear regression techniques.Because site index can influence height, alternative equations includingtransformations of site index as an Independent variable also are presentedfor 6 of these 16 species.As an affirmative action institution that complies with Section504 of the Rehabilitation Act of 1973, Oregon State Universitysupports equal educational and employment opportunity withoutregard to age, sex, race, creed, national origin, handicap, maritalti status, or religion.Forest Research LaboratoryCollege of ForestryOregon State UniversityCorvallis, OR 97331-5704°State"UniversityiNonProfit OrgUS PostagePAIDCorvallis. OR 97331Permit No 2001I4Address Correction Requested

Height-diameter equations for sixteen tree species in the central western Willamette Valley of Oregon

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Height-diameter equations for sixteen tree species in the central western Willamette Valley of Oregon

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