Objective  In order to analyze the influence of thinning on the number of branches for Pinus koraiensis plantation, this study constructed a biological mathematic model based on the thinning effect, and provided a theoretical basis for developing a scientific and reasonable thinning program.

  Method  Based on the data of 4 370 branches from 49 sample trees in Pinus koraiensis plantation in Linkou and Dongjingcheng Forestry Bureau of Heilongjiang Province of northeastern China, this study established a single-level nonlinear mixed effect model of branch density with thinning effects using nlme package of R. The converged models were then evaluated by adjusted coefficient of determination ( R_\rma^2 ), Akaike information criterion AIC, Bayesian information criterion (BIC), log likelihood and likelihood ratio test (LRT).

  Result  When site index and tree size were similar, branch density increased with the increase of thinning intensity and crown length. When thinning intensity and tree size were similar, branch density increased with the increase of site index and crown length. However, when thinning density and site index were similar, branch density was negatively correlated with DBH. Nonlinear mixed effect model with plot effect had higher fitting precision than that with tree effect and corresponding fixed effect model. Finally, the nonlinear mixed model with five random coefficients, including DINC (depth into crown), lnRDINC (natural logarithm of relative depth into crown), RDINC² (square of relative depth into crown), DBH and TI/TA (thinning intensity over thinning age) was selected as the most optimized model for predicting branch density, whose R_\rma^2 was 0.825 7 and RMSE was 2.171 4.

  Conclusion  The optimal nonlinear mixed effect model with thinning effect not only has higher precision, but also more accurately reflects the effect of thinning on tree branches than other models.