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Individual trees are characterized by various sizes and forms, such as diameter at breast height, total height (H), height to crown base (HCB), crown length (CL), crown width, and crown and stem forms. Tree characteristics are strongly related to each other, and studying their relationships is very important. The knowledge of the compatibility and additivity properties of the major tree characteristics, such as H, CL, and HCB, is essential for informed decision-making in forestry. H can be used to represent site quality and CL represents biomass and photosynthesis of crown, which is the performance of individual tree vigor and light interception, and the longer the crown length (or shorter HCB) is, the more vigorous the tree would be. However, none of the studies have uncovered their inherent relationships quantitatively. This study attempts to explore such relationships through the application of appropriate modeling approaches. We applied seemingly unrelated regression, such as nonlinear seemingly unrelated regression (NSUR), which is commonly used for exploring the compatibility and additivity properties of the variables, for the proposes. The NSUR involves the variance and covariance matrices of the sub-models that are used for the interpretation of the correlations among the variables of interest. The data set acquired from Mongolian oak forest and spruce-fir forest in the Jingouling forest farm of the Wangqing Forest Bureau in the Northeast of China were used to construct two types of model systems: a compatible model system (the model system of H, CL, and HCB can be estimated simultaneously) and an additive model system (the sum of HCB and CL is H, the form of the H sub-model equals the sum of the HCB and CL sub-models) from the individual models of H, CL, and HCB. Among the various tree-level and stand-level variables evaluated, D (diameter at breast), Dg (quadratic mean diameter), DT (dominant diameter), CW (crown width), SDI (stand density index), and BAS (basal area of stand) contributed significantly highly to the variations of the response of the variables of interest in the model systems. Modeling results showed the existence of the compatibility and additivity of H, CL, and HCB simultaneously. The additive model system exhibited better fitting performance on H and HCB but poorer fitting on CL compared with the simultaneous model system, indicating that the performance of the additive model system could be higher than that of the simultaneous model system. Model tests against the validation data set also confirmed such results. This study contributes a novel approach to solving the compatibility and additivity of the problems of H, CL, and HCB models through the application of the robust estimating method, NSUR. The results and algorithm presented will be useful for constructing similar compatible and additive model systems of multiple tree-level models for other tree species.

Simultaneous Compatible System of Models of Height, Crown Length, and Height to Crown Base for Natural Secondary Forests of Northeast China