United States
Department of
Agriculture
Forest Service
Pacific Southwest
Research Station
Research Paper
PSW-RP-227-Web

Crown Area Equations for 13 Species of Trees and Shrubs in Northern California and Southwestern Oregon

Fabian C.C. Uzoh    Martin W. Ritchie

Cover


July 1996

Pacific Southwest Research Station

Forest Service
US Department of Agriculture



Abstract:

Uzoh, Fabian C.C.; Ritchie, Martin W. 1996. Crown area equations for 13 species of trees and shrubs in northern California and southwestern Oregon. Res. Paper PSW-RP-227-Web. Albany, CA: U.S. Department of Agriculture, Forest Service, Pacific Southwest Research Station; 13 p.


  The equations presented predict crown area for 13 species of trees and shrubs which may be found growing in competition with commercial conifers during early stages of stand development. The equations express crown area as a function of basal area and height. Parameters were estimated for each species individually using weighted nonlinear least square regression.

Retrieval terms: growth and yield, simulators


Authors

Fabian C.C. Uzoh and Martin W. Ritchie are Research Forester and Research Statistician, respectively, in the Station's Conifer Silviculture Research unit, 2400 Washington Avenue, Redding, CA 96001.


Acknowledgments

The data used in this analysis were provided by Dr. Robert J. Laacke of Pacific Southwest Research Station, 2400 Washington Avenue, Redding, CA 96001.


Contents



In Brief

Introduction

Methods
    Measurements

Analysis and Results

Concluding Discussion

References



In Brief...

Uzoh, Fabian C.C.; Ritchie, Martin W. 1996. Crown area equations for 13 species of trees and shrubs in northern California and southwestern Oregon. Res. Paper PSW-RP-227-Web. Albany, CA: Pacific Southwest Research Station, Forest Service, U.S. Department of Agriculture; 13 p.

Retrieval Terms: growth and yield, simulators


Growth and yield simulators are valuable tools in forest management decision making. A number of simulators have been developed to meet current management challenges. However, little attention has been given to the impact of brush and non-commercial hardwood trees on stand dynamics. This presents a serious problem when attempting to predict the early stages of stand development where shrubs and sprouting hardwood trees may have a significant impact on growth of commercial species. Crown area may be useful as an index of the competition level. One means of estimating crown area is to predict it as a function of variables which may be readily available in a simulator, such as height or diameter. In this paper, we present crown area equations for 13 species of plants common to southwestern Oregon and northern California. Crown area is expressed as a function of stem basal area and height for each plant. Parameters were estimated for each species individually using weighted nonlinear least square regression. For the purpose of this study, crown area is defined as the area of a vertical projection of the crown to a horizontal plane.


Introduction

Growth and yield models are becoming one of the primary tools aiding forest managers in decision making. Traditionally, modeling efforts have focused on merchantable species. In the western United States, these are primarily coniferous trees. As a result, very little modeling work has been done on hardwood trees or shrubs.

   There is increasing interest in simulators which are capable of predicting growth for early seral stages of stand development. These simulators must address the influence of shrubs and hardwood trees. One such model is SYSTUM-1 (Ritchie and Powers 1993). In SYSTUM-1, competing vegetation is quantified as an aggregate percent cover and height by species on each plot. However, SYSTUM-1 is not responsive to changes in species composition on tree growth. Furthermore, growth functions for percent cover and height are not effective in representing response to release treatments. In order to address this problem, stem growth models are being developed for shrubs and hardwood trees in which crown area is expressed as a function of crown volume. This work is part of a larger effort to develop a suite of functions which will allow simulators to better respond to questions relating to the impact of non-commercial species on growth of commercial conifers.

   The objective of this study is to develop equations to predict crown area as a function of plant height and basal diameter. Crown area and crown cover values have been used in studies of biomass production (Martin and others 1981; Mitchell and others 1987) as well as in predicting the competitive effects of shrubs on tree growth (e.g. Dahms 1950; Oliver 1979). A crown area function may help existing simulators such as SYSTUM-1 provide more reliable predictions of shrub cover over time. For the purpose of this study, crown area is defined as the area of a vertical projection of the crown to a horizontal plane.

   This paper presents equations that predict crown area for the following 13 species of trees and shrubs: Vine maple (Acer circinatum [Pursh]), Manzanita (green leaf) (Arctostaphylos patula [Adams]), Golden chinkapin (Castanopsis chrysophylla [Dougl.] A.DC.), Whitethorn (mountain) (Ceanothus cordulatus), Deerbrush (Ceanothus integerrimus [Hook. & Arn.]), Snowbrush ( Ceanothus velutinus [Dougl.]), Tanoak (Lithocarpus densiflorus [Hook and Arn.] Rehd.), Black oak (Quercus kelloggii [Newb.]), Pacific madrone (Arbutus menziesii [Pursh]), Bitter cherry (Prunus emarginata [Dougl.]), Pacific rhododendron (Rhododendron macrophyllum [D. DON.]), Huckleberry (bush) (Vaccinium ovalifolium [Smith]), and Bigleaf maple ( Acer macrophyllum [Pursh]).


Methods

The data for this study were collected from temporary plots in 29 plantations in northern California and southwestern Oregon during the summers of 1987, 1988, and 1989 (fig. 1). The plantations were between 25 and 45 acres in size. On each plantation, 30 plots (15 non-tree-centered and 15 tree-centered plots) 2.6 meters in radius were established systematically at 40-meter intervals along a continuous transect that changed direction every 200 meters. The choice of the initial starting point at each plantation was made by selection from a random number table. Even digits (0,2,4, etc.) designated a tree-centered plot, odd digits a vegetation plot.

   When a tree-centered plot was chosen, the plot center was moved from the transect center line to coincide with the location of the nearest seedling. However, if there were no seedlings within 2.6 meters of the 40-meter mark on the transect, the plot number was recorded, no other data taken, and the transect continued. Plot type refers to whether the plot was tree-centered or not. Data collected from the two types of plots were the same. A plant was considered to be in the plot if the center of the plant where it emerged from the ground was within 2.6 meters of plot center. For all plants in the plot, the data were collected from the entire plant, including those parts that extended outside the plot. Parts of a plant that did not emerge from the soil surface within the plot were not measured. The only exception to this rule was during estimation of percent cover (percent cover is the ocular estimation of percent of grasses and forbs at each plot). The majority of the plants measured were shrub clumps often from multi-stemmed individuals.

Measurements

The following were recorded for each woody plant in a plot:

Equation page 3

Plant basal area (BA) in square centimeters is calculated as:

Equation page 3

where:

ns = number of stems per plant.


Analysis and Results

We assumed that there is no plot effect, and we treated each individual plant as if it were drawn at random from all possible plants. Two model forms were considered:

CA=a ×(1-EXP(HT × b)) +error

[1]

CA=(a × BAb) × (1-EXP(c×HT)) + error

[2]

where a, b, and c = regression coefficients. Because of the availability of height measurements, and the difficulty of measuring basal area in shrubs, a model using only height may be more desirable. Of these two, Eq. [2] had the lower mean square error across species. Further, unweighted fit R2 values for equation [1] ranged from .0603 to .6802 while R2 values for Eq. [2] ranged from .4288 to .9237. Also, a more generalized method of model comparison (Anderson-Sprecher 1994) was used to compare the two equations. TheirGR2 parameter is an index of how much the full model (Eq. #2) improves over the reduced model(Eq. #1). The values of GR2 are presented in Table 1. Given the values of R2 and GR2, Eq. [2] was chosen. The estimate of the coefficient of determination, R2, was calculated using the formula: R2 = 1 - [SSE/CTSS], where SSE = Sum-of-squares error about the equation; and CTSS = Corrected total sum of squares. The estimate of was GR2 calculated using the formula: GR2 = 1 - [RSS(full)/RSS(reduced)], where RSS(full) = Residual sum of squares about Eq. [2], and RSS(reduced) = Residual sum of squares about Eq. [1]. This approach provides a generalized R2 value which implies a comparison with the reduced model CA= a + error (Anderson-Sprecher 1994).

Crown area is modeled as a function of stem basal area and height. Preliminary plotting of crown area against the independent variables revealed the following:

   Initially, parameters were estimated for all 13 species using unweighted nonlinear least squares regression. Plotting of the residuals indicated that the values of the residuals were increasing over the estimate. To improve parameter estimates, we employed an iteratively reweighted nonlinear least squares fit. We tried two weights to homogenize variance. We weighted by [1/expected(CA)] 1 /2   and 1/expected(CA). These two weights correspond to assumptions that variance of residuals increased with expected(CA) and square of expected(CA) respectively. Residual plots and Furnival's (1961) index were used to compare unweighted regression with each of the weighted regressions to determine which weight best met the assumptions of homogeneity of variance and normality of the error term. In all cases, the weighted regressions provided the lowest index of fit. The weight of 1/expected(CA)  provided what appeared to be the best fit for 7 of the 13 species. In most instances, convergence occurred before six iterations. Tables 2  and 3  present the regression coefficients and associated statistics for the modeling data set for Eq. [2]. Figures 2-14  show resulting functions over basal area for five levels of height. Comparison of these plots with those from unweighted fits did not reveal any dramatic statistically significant difference. The SAS (SAS Institute Inc. 1987) statistical software package was used to estimate the regression parameters.


Concluding Discussion

These equations are intended primarily for the purpose of predicting missing data values for data describing the structure of young plantations. Crown area is a variable which may be useful in describing levels of competition in such stands (e.g., Tappeiner and others 1984). Since height growth is easily measured on shrubs and hardwoods in young stands, a crown area using only height as a predictor is preferable. However, we found substantial improvements using total basal area and height in the model. We believe the models using height alone were inadequate for predicting crown area; therefore we did not present the results of these fits.

   In the chosen model, crown area increases monotonically with increasing basal area and height. This may provide for reasonable behavior beyond the range of the data in our sample. However, we have no basis for assessing the robustness of these equations to extrapolation. Although the model chosen contains an asymptote which is conditional on the value of BA, for most of the species, the sampled data range is well below any implied asymptote. Therefore no conclusions may be drawn about maximum crown area from these equations.

   The species depicted in figures 2-14  show substantial variability. The variability between species is to be expected given the wide range in form of these species. The tanoak and madrone models are very similar within the ranges of data common to both species. However, the tanoak sampled in this study covered a much wider range in basal area than was found among the sampled madrone.

   The relationship between crown area and height and basal area may be influenced by origin of the plant. Among these plantations, shrubs and hardwoods have developed both from seed and sprouts after fire or release treatment, and we make no effort here to distinguish between the two. Parameter estimates may be influenced by the particular mix of seed and sprout origin of individuals in the data set. Model performance may be enhanced if this factor is explicitly included in the model.



Figures 2-14

Figure 2--Plot of fitted equation for vine maple--crown area as a function of height (m) and basal area (cm2).

Figure 3--Plot of fitted equation for manzanita--crown area as a function of height (m) and basal area (cm2).


Figure 4--Plot of fitted equation for chinkapin--crown area as a function of height (m) and basal area (cm2).

Figure 5--Plot of fitted equation for whitethorn--crown area as a function of height (m) and basal area (cm2).


Figure 6--Plot of fitted equation for deerbrush--crown area as a function of height (m) and basal area (cm2).

Figure 7--Plot of fitted equation for snowbrush--crown area as a function of height (m) and basal area (cm2).


Figure 8--Plot of fitted equation for tanoak--crown area as a function of height (m) and basal area (cm2).

Figure 9--Plot of fitted equation for black oak--crown area as a function of height (m) and basal area (cm2).


Figure 10--Plot of fitted equation for madrone--crown area as a function of height (m) and basal area (cm2).

Figure 11--Plot of fitted equation for bitter cherry--crown area as a function of height (m) and basal area (cm2).


Figure 12--Plot of fitted equation for rhododendron--crown area as a function of height (m) and basal area (cm2).

Figure 13--Plot of fitted equation for huckleberry--crown area as a function of height (m) and basal area (cm2).

Figure 14--Plot of fitted equation for bigleaf maple--crown area as a function of height (m) and basal area (cm2).



References

Anderson-Sprecher, R. 1994. Model comparisons and R2. American Statistician 48: 113-117.

Dahms, Walter G. 1950. The effects of manzanita and snowbrush competition on ponderosa pine reproduction. Res. Note 65. Portland, OR: Pacific Northwest Forest and Range Experiment Station, Forest Service, U.S. Department of Agriculture; 3 p.

Furnival, George M. 1961. An index for comparing equations used in constructing volume tables. Forest Science 7: 337-341.

Martin, Robert E.; Frewing, David W.; McClanahan, James L. 1981. Average biomass of our northern shrubs by fuel size class and crown cover. Res. Note PNW-374. Portland, OR: Pacific Northwest Forest and Range Experiment Station, Forest Service, U.S. Department of Agriculture; 6 p.

Mitchell, J.E.; Bartling, P.N.S.; O'Brien, R. 1987. Understory cover-biomass relationships in the front range ponderosa pine zone. Res. Note RM-471. Fort Collins, CO: Rocky Mountain Forest and Range Experiment Station, Forest Service, U.S. Department of Agriculture; 5 p.

Oliver, William W. 1979. Early response of ponderosa pine to spacing and brush: observations on a 12-year-old plantation. Res. Note PSW-341. Berkeley, CA: Pacific Southwest Forest and Range Experiment Station, Forest Service, U.S. Department of Agriculture; 7 p.

Ritchie, Martin W.; Powers, Robert F. 1993. User's guide for SYSTUM-1 (Version 2.0): A simulator of growth trends in young stands under management in California and Oregon. Gen. Tech. Rep. PSW-GTR-147. Albany, CA: Pacific Southwest Research Station, Forest Service, U.S. Department of Agriculture; 45 p.

SAS Institute Inc. 1987. Guide for personal computers, version 6 Edition. Cary, NC: SAS Institute Inc.; 1028 p.

Tappeiner, John C.; Harrington, Timothy B.; Walstad, John D. 1984. Predicting recovery of tanoak (Lithocarpus desiflorus) and Pacific madrone (Arbutus menziesii) after cutting or burning. Weed Science 32: 413-417.