lower portions of trees were measured, but most were 

 estimated by sight. Cubic foot volume was computed for 

 each segment using Huber's log formula (Husch and 

 others 1982, p. 101). For each tree, segments that in- 

 cluded wood and bark of all stems and branches larger 

 than 1.5 inches in diameter were summed into a gross 

 cubic foot volume. Both live and dead wood were 

 included. 



Other tree variables measured for volume equation 

 development were total height, DRC, and number of basal 

 stems. For trees forking at the root collar, an equivalent 

 diameter (EDRC) was calculated for use in place of DRC: 



EDRC = 



i=l 



(1) 



where 



n = number of basal stems 1.5 inches or larger 



D = basal diameter of each stem, 

 i 



Species sampled, followed in parentheses by numbers 

 sampled, were Juniperus osteosperma (Torr.) Little (266), 

 J. monosperma (Engelm.) Sarg. (169), J. deppeana Steud. 

 (95), J. erythrocarpa Cory (15), Pinus edulis Engelm. 

 (123), P. cembroides Zucc. (72), P. edulis var. fallax Little 

 (31), Prosopis velutina Woot. (290), Acacia greggii Gray 

 (19), Olneya tesota Gray (12), Quercus emoryi Torr. (85), 

 Q. arizonica Sarg. (37), Q. hypoleucoides A. Camus (23), 

 Q. oblongifolia Torr. (17), Q. gambelii Nutt. (5), and 183 

 combined Quercus hybrids, Q. turbinella Greene, 

 Q. chrysolepis Liebm., and perhaps a few other evergreen 

 oaks. For initial analysis, species were combined by ge- 

 nus, except for Prosopis, Acacia, and Olneya, which were 

 combined into a mesquite group (table 1). 



MODELING 



Volume was compared to the combination variable, 

 DRC or EDRC squared times height divided by 1,000 

 (DSQH), and other data characteristics through graphing. 

 This was done to examine the volume data for differences 

 among species and to examine the volume-to-DSQH 

 relationship. 



Little evidence was found to separate the data beyond 

 juniper, oak, pinyon, and mesquite species groups. Either 

 a species was represented by too few trees or species vol- 

 ume variation within a group was similar for all species. 

 Visual volume estimation was considered a possible 

 source of variation masking differences between species. 

 Even though visual volume estimation was shown 

 adequate for constructing volume equations (Born and 

 Chojnacky 1985), no validation checks were available to 

 confirm the reliability of the Arizona data. Some mes- 

 quite volume data from Pima and Santa Cruz Counties 

 showed less variation in data estimated by one individual 

 than in data estimated by several field crews combined. 

 This probably suggests that visual volume estimation 

 error should be considered if there is a need to identify 

 precise volume relationships between individual species. 



Graphs comparing the volume-to-DSQH relationship for 

 all species groups supported the idea that volume can be 



adequately predicted from DSQH, even though the model 

 relationship tends to change for larger trees. The graphs 

 also showed the volume relationship differed slightly be- 

 tween single-stem and multiple-stem trees. 



The point of change in the volume relationship between 

 small and large trees was identified from graphs. Trees 

 for each species group were ranked from smallest to larg- 

 est, and the point of change was observed between the 

 90th and 95th percentile, corresponding to DSQH values 

 of 6, 4, 3, and 2 for juniper, oak, pinyon, and mesquite, 

 respectively. 



An analysis of variance to test for differences within 

 species groups was done to compare data from northern 

 Arizona counties with data from southern Arizona counties 

 (Chojnacky 1985, p. 5; Graybill 1976, p. 247). All four 

 species groups showed no significant difference between 

 northern and southern Arizona data (F-test significance 

 levels were 0.19 or greater). 



Based on the results of the graphical analysis and analy- 

 sis of variance, volume equations were constructed for 

 seven categories: 



1 . Single-stem juniper 



2. Multiple-stem juniper 



3. Single-stem oak 



4. Multiple-stem oak 



5. Single -stem mesquite 



6. Multiple-stem mesquite 



7. Single-stem pinyon (with 11 multiple-stem trees 

 included). 



The model was broken into two parts to allow for differ- 

 ences between the largest trees and the rest: 



V = 



[ P 3 + (yr-H P/X forX>X Q 



(2) 



where 



V = gross cubic foot volume of wood and bark from all 

 stems and branches larger than 1.5 inches in diameter 



X = DRC or EDRC squared times height divided by 1,000 



P t - = parameters estimated from data 



X o = 



6 for juniper (roughly a 20-inch DRC) 



4 for oak (roughly a 16-inch DRC) 



3 for pinyon (roughly a 12-inch DRC) 



2 for mesquite (roughly a 12-inch DRC) 



The two-part volume model was conditioned to be both 

 smooth and continuous at the point where the two equa- 

 tions meet. This was done by imposing two restrictions on 

 the model: 



h = Po + Vo 2 <» 

 P 4 = -2P 2 X Q 3 (4) 



The restrictions were obtained by equating the two parts 

 of equation 2 for X equal toX Q , and by equating the first 

 derivative of the two parts of equation 2 at the point X 

 equal to X Q . Parameters for the two-part model (eq. 2) 

 were determined using weighted regression with DSQH 



3 



