relation of height, diameter, and diameter increment. These effects could be incorpo- 

 rated quite readily by multiplying each independent variable by a coefficient to be 

 estimated, and by introducing a constant term for each species. In the logarithmic 

 form, the possibility of zero values for A£> must be considered. Measurements of diam- 

 eter growth are usually recorded only to the nearest 1/20 inch. By shifting the entire 

 scale up 1/20 inch, zero values can be accommodated without distorting the overall 

 relation . 



Thus, the logarithmic form of the differential model is: 



ln(Aff) = Q\ + e 2 ln(AZ) + .05) + o 3 In (2?) + c h ln(fl) (5) 



t 



For the 909 trees from the northern Idaho forests, the following coefficients for 

 this model gave a better index of fit (±2.73 feet) than the differential model with the 

 loss of fewer degrees of freedom: 







4448 



Western white pine 







1258 



Western larch 





(1 



1424 



Douglas- fir 







2939 



Grand fir 







1508 



Western hemlock 







9357 



Western redcedar 







0706 



Lodgepole pine 





I! 



2221 



Engelmann spruce 







8355 



Subalpine fir 







4380 



Ponderosa pine 



°2 = 



+0. 



37401 



In (AD + 0.05) 



°l = 



-0. 



29805 



ln(ZT) 



c h = 



-0. 



13170 



In (H) 



s 2 = 



0. 



1775 





When the same model was applied to the 265 lodgepole pine and Douglas-fir trees 

 from the Lewis and Clark National Forest, the values of the coefficients were quite 

 different from those above. The estimated residual variance for these data was 

 s 2 = 0.1630. The difference between these two populations is apparent in the following 

 averages : 



Northern Idaho Forests Lewis and Clark N.F. 



AH 6.5 1.4 



AO 1.8 0.44 



Differences in growth rates correspond to marked differences in habitat for those 

 species that are common to the two geographic sources of data. As a consequence of the 

 differing habitats, the Lewis and Clark trees would show a much lower average site index 

 than the northern Idaho trees. Furthermore, the Lewis and Clark trees are generally 

 older. To accommodate these differences, data from the two sources were merged and the 

 differential model modified to permit coefficients to vary with habitat as well as 

 species. If all four coefficients were unique with respect to species and habitats, 

 then there would be 204 coefficients to be estimated by a separate regression solution 

 of equation (5) for each of the 51 entries in table 1. Instead, the four coefficients 

 in the logarithmic form of the differential model were varied according to species and 



12 



