The segment cubic volumes were then summarized by diameter class to provide estimates of 

 cubic volume for each tree by six different top diameters--l in through 11+ in (2.54 cm- 

 27.94 cm) by 2-in (approximately 5 cm) odd increments. Cubic volume, so calculated, was 

 used as the dependent variable for regression analyses. 



A combinatorial screening approach (Grosenbaugh 1967) was used initially to deter- 

 mine which combination of pertinent transforms of all independent variables gave the best 

 estimate of cubic volume. The results indicated basal diameter and total height to be 

 the best predictors of cubic volume for pinyon. Basal diameter, height, and number of 

 forks provided the "best" correlation with juniper volume. Substituting number of stems 

 for number of forks gave a slightly smaller correlation; however, number of stems was 

 selected for the final juniper volume model because number of stems is more easily and 

 accurately determined in the field. 



After these independent variables were selected, the data were summarized by 

 species for each minimum top diameter limit. For pinyon, mean cubic volume was computed 

 for each 2-in (approximately 5 cm) basal diameter class and each 5-ft (1.5 m) height 

 class. For juniper, mean cubic volume was computed for each 2-in (5 cm) basal diameter 

 class, 5-ft (1.5 m) height class, and number-of- stems class. These mean cubic volumes 

 were then used to develop a refined volume model that was finally fitted to all indi- 

 vidual tree volumes by linear least squares techniques. 



The following constraints were applied to the voliome models: 



1. Volume should increase over height and must pass through zero cubic volume 

 when height equals zero. 



2. Volume should increase over basal diameter and must pass through zero cubic 

 volume when basal diameter equals minimum top diameter limit. 



3. Volume must approach zero cubic volume as number of stems increases. 



4. Volume must decrease with increasing minimum top diameter limit. 



5. The minimum top diameter components must not cross. 



6. The number of stem components of model must not cross. 



An initial plotting of the data indicated volume increased linearly over height 

 and was concave upward over basal diameter. 



The first constraint is satisfied by a model of the form Y = bH where b is ex- 

 pressed as a function of basal diameter, number of stems, and minimum top diameter 

 limit. Minimum top diameter limits of 9 in (22.86 cm) and 11 in (27.94 cm) were dropped 

 from the analysis because inadequate sample sizes resulted in inconsistent trends. The 

 final model was then developed using minimum top diameter limits of 1, 3, 5, and 7 in 

 (2.54, 7.62, 12.70, and 17.78 cm). 



The first step in the model building process was to fit a weighted linear equation 

 of the basic model form through the origin with height as the independent variable and 

 cubic volume as the dependent variable. This was done for each species, 2-in (5 cm) 

 basal diameter class, minimum top diameter limit, and in the case of juniper, each 

 number-of-stem class up to six. Beyond six stems, the data base was insufficient to 

 determine consistent trends. The weights used were the number of observations contrib- 

 uting to each mean volume. 



2 



