the effects of site, stand density, individual tree vigor, and more. But if we wished 

 to model the effects of thinning or partial cutting, then the prognoses would be 

 inadequate because the changes in stand density would not be reflected in changes in 

 growth rates. The past growth variable would be an alias for stand density effects. 

 Similar difficulties could arise from the combination of age and size in the predic- 

 tion because size divided by age is, in effect, a measure of past growth. 



The key growth function predicts the rate of increase of tree d.b.h. The depen- 

 dent variable is the logarithm of the annual increase in the square of d.b.h. in 

 inches; thus, this variable is equivalent to the logarithm of the basal area increase. 

 Basal area was selected because its increase is most frequently linear with time. This 

 linearity facilitates projection for intervals different from the growth interval over 

 which the parameters of the model were estimated. If basal area increase is measured 

 without bark, then the ratio of basal area outside bark to basal area inside bark is 

 used to convert the increment to outside-bark measure. 



The logarithmic transformation is used here for two reasons: First, diameter 

 growth rate distributions are bounded by zero at the lower end (ignoring the effects 

 of bark sloughing) and so tend to be quite skewed; hence, the arithmetic mean is an 

 inefficient estimator. Second, the variability of diameter growth rates about their 

 mean tends to increase as the mean increases . The logarithmic transformation in most 

 cases has resulted in a uniform variance. The method of Oldham (1965) is used to 

 estimate the arithmetic mean of the diameter growth from the logarithmic model. A 

 further refinement developed by Bradu and Mundlak (1970) was considered unnecessary 

 because the standard errors of estimate are usually small enough that the differences 

 between the two methods are trivial. 



Predictor variables that are available include the site and tree characters listed 

 previously. Additional variables that measure stocking or relative stand density such 

 as crown competition factor, basal area, or stand density index can be computed from 

 the distribution of the tree diameters. Variables that measure the competitive rela- 

 tions between trees in the stand include the percentile in the basal area distribution 

 or the ratio of tree d.b.h. to mean stand diameter (Appendix I). In addition, predic- 

 tions for rates computed earlier in the sequence of calculations can be used as 

 predictors. These models are summarized ais follows: 



Tree gvoidth aomponents Predictor vari-ables Data source 



Annual basal area D.b.h., relative stand density, Increment cores, 



increment (b.a.i.) site, elevation, habitat type, remeasured plots 



percentile in basal area 



distribution, crown ratio 



Height increment 



Radial increment, habitat 

 type, d.b.h., height 



Stem analyses 



Crown dimensions 



Relative density, percentile 

 in basal area distribution, 

 d.b.h. 



Temporary plots 



Bark ratio 



Mortality rates 



Same as b.a.i, 



Same as b.a.i. and radial 

 increment plus pest popula- 

 tion models where applicable 



Temporary plots or 

 tree samples 



Remeasured plots, 

 "last n years 

 mortality", "years 

 since death" 

 (truncated) 



9 



