For large samples, Stage randomly picks a deviate from the normal distribution of the 

 residuals about the log of basal area growth regression equation and adds it to the estimate 

 of log of basal area growth. The underlying assumption is that, with a large sample, the 

 effects of the individual random deviations will average out over the stand. 



For small samples, each sample tree record is divided into three sample tree records. 

 The number of trees in each record is a fixed proportion of the original number of trees 

 represented by the old sample tree record. The proportion breakdown Stage used was 15, 60, 

 and 25 percent, based on previous findings that an average stand had 15 percent suppressed 

 trees and 25 percent dominant trees. ^ Each of the new sample tree records is assigned a 

 growth rate computed by taking the average growth rate (as predicted by the log basal area 

 growth equations) and adding a random component. For the 25 percent dominant trees, the 

 random component is the expected residual value of the largest 25 percent of the normally 

 distributed residuals about regression; for the 15 percent suppressed trees, it is the expected 

 residual value of the lowest 15 percent of the normally distributed residuals, and similarly 

 for the middle trees. As a result, the weighted log of basal area growth still sums to the 

 average log of basal area growth, as predicted by the equation. At each simulation period, 

 the sample tree records are split again until enough sample trees exist so that the first 

 method can be used. 



Certain aspects of the second method seem applicable to this study. Within a diameter 

 class, individual tree growth could be quite variable. By using only average diameter growth, 

 all trees in the class will be assigned the same growth rate and, therefore, advancement to 

 larger diameter classes will be identical. If, however, the number of trees in a diameter 

 class were divided into thirds (each third representing the fast, slow, and moderate growers), 

 then a separate growth rate could be assigned each third in the same fashion as used by Stage. 

 If the three growth rates differed enough, this would then allow the trees in a diameter class 

 to move into a wider range of larger diameter classes. If the distribution of residuals 

 remains constant over time, this approach should provide a more realistic estimate of the 

 number of trees changing diameter classes. Thirds were used in this study instead of Stage's 

 proportions because the proportion of suppressed and dominant trees for southwestern ponderosa 

 pine is unknown. 



Definition of Independent and Dependent Variables 



The previously discussed factors influencing diameter growth include diameter class size, 

 productivity, rainfall, total stand density, vigor, and position of the diameter class within 

 the stand. 



Over diameter class size (D) , it is expected that basal area growth will climb to some 

 peak value and then drop off. Where and at what magnitude the peak will occur is unknown. 

 One function that would allow a wide range of peaking forms is the Weibull function. The 

 Weibull function can be expressed as (Bailey and Dell 1973): 



or 



, where b and c are parameters 



^Based on a presentation given by Albert R. Stage, principal mensurationist with the 

 Intermountain Forest and Range Experiment Station, USDA Forest Service, in a spring 1976 

 graduate seminar at the University of Washington. 



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