Bivariate distribution functions for predicting twig leaf area within hybrid spruce crowns

Abstract

The vertical and horizontal distribution of leaf area per centimetre of twig (APCM) for hybrid spruce (Picea engelmannii Parry ex Engelm. × Picea glauca (Moench) Voss × Picea sitchensis (Bong.) Carrière) tree crowns was modelled using bivariate Weibull and beta distribution functions. Horizontal position was represented by relative position on the first-order branch. Vertical position was based on branch position from the tree apex relative to tree height, since the base of the live crown is often difficult to locate. Sample APCM measures were obtained using systematic sampling of 12 tree crowns taken from stands at three developmental stages (20, 60, and 140 years of age). For comparison, univariate Weibull and beta distribution functions using only vertical distribution were also fitted. Generally, APCM decreased from the tree apex downward and from the branch tips toward the stem, although variation in the values was quite high. Trees from the middle stand age (60 years) had the highest average APCM values, followed by the smallest, youngest trees (stand age 20), and the lowest values were found for the largest trees in the oldest stand (140 years). As anticipated, the bivariate Weibull and beta distribution functions resulted in more precise representations of APCM within tree crowns than the univariate Weibull and beta distribution functions, although the improvements were minimal. Results were generally poorer for trees from the oldest stand. These functions could be used to evaluate other variables distributed over the tree crown, such as specific leaf area. The resulting models from this study were used to reconstruct the entire crown of every sample tree for conducting sampling simulations.