Research Highlights: this study developed additive biomass equations respectively from nonlinear regression (NLR) on original data and linear regression (LR) on a log-transformed scale by nonlinear seemingly unrelated regression (NSUR). To choose appropriate regression form, the error structures (additive vs. multiplicative) of compatible biomass equations were determined on the use of the multivariate likelihood function which extended the method of likelihood analysis to the general occasion of a contemporaneously correlated set of equations. Background and Objectives: both NLR and LR could yield the expected predictions for allometric scaling relationship. In recent studies, there are vigorous debates on which regression (NLR or LR) should apply. The main aim of this paper is to analyze the error structure of a compatible system of biomass equations to choose more appropriate regression. Materials and Methods: based on biomass data of 270 trees for three tree species, additive biomass equations were developed respectively for NLR and LR by NSUR. Multivariate likelihood functions were computed to determine the error structure based on the multivariate probability density function. The anti-log correction factor which kept the additive property was obtained separately using the arithmetic and weighted average of basic correction factors from each equation to assess two model specifications on the comparably original scale. Results: the assumption of additive error structure was well favored for an additive system of three species based on the joint likelihood function. However, the error structure of each component equation calculated from the conditional likelihood function for compatible equations might be different. The performance of additive equations corrected by a weighted average of basic correction factor from each component equation performed better than that of the arithmetic average and held good property of compatibility after corrected. Conclusions: NLR provided a better fit for additive biomass equations of three tree species. Additive equations which confirmed the responding assumption of error structure performed better. The joint likelihood function on the use of the multivariate likelihood function could be used to analyze the error structure of the additive system which was a result of a tradeoff for each component equation. Based on the average of correction factors from each component equation to correct the bias of additive equations was feasible for the hold of additive property, which might lead to a poor correction effect for some component equation.

In the subtropical mountainous area of Huangshan, China, 81 Torreya grandis were selected for research on growth and element distribution, with the aim of providing empirical and theoretical support for the promotion and management of T. grandis. Seven growth equations were constructed for tree height (H) and diameter at breast height (DBH) in age levels 40–2000. The most suitable Korf equation was selected by comparing the R-squared values. With R-squared values of 0.93 and 0.80, the equations H = 33.98 × Exp(−53.56/age^0.84) and DBH = 158.48 × Exp(−21.41/age^0.55) fit T. grandis growth best, respectively. The results show that T. grandis growth was relatively rapid before the age of 45 and then slowed. Among the carbon (C), nitrogen (N), and phosphorus (P) elements found in different T. grandis organs of the same age, the roots had the lowest C content, the leaves had the highest N content, and the branches had the lowest P content. The roots had the highest C:N and C:P, and the leaves had the highest N:P. The amount of N in the leaves and P in the branches both increased significantly with age, and the N:P ratio indicates that considering only one organ may not accurately indicate T. grandis nutrient limitations.

This study reviewed published biomass equations for Larix species in the literature and developed additive biomass equations (by nonlinear seemingly unrelated regression known as NSUR) and independent equations, respectively, based on DBH-only and DBH-H variables with the biomass data of 150 trees sampled in northeastern China. The applicability of fitted and published equations was comprehensively assessed with respect to Larix species, their provinces and origins when applied in northeastern China. The results showed that equations derived from the local study area did not always perform better than the published biomass equations in the literature. Tree-level equations performed slightly broader than stand-level equations. Nonlinear regression could provide higher applicability due to correction factors in the linear regression. Adding tree height as the second variable improved the applicability of stem and aboveground published equations. The applicability of original published equations was better for northeastern China for sites of similar latitude. When using published biomass equations, care should be taken to assess the uncertainty and applicability of biomass equations, which is important for the sustainable management of forested resources. Published information on biomass equations is especially necessary for forestry organizations that do not have the opportunity to develop their own equations.