# Using Linear Mixed-Effects Models with Quantile Regression to Simulate the Crown Profile of Planted Pinus sylvestris var. Mongolica Trees

Department of Forest Management, School of Forestry, Northeast Forestry University, Harbin 150040, China

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Received: 17 October 2017 / Revised: 9 November 2017 / Accepted: 14 November 2017 / Published: 17 November 2017

(This article belongs to the Special Issue Successional Dynamics of Forest Structure and Function)

Crown profile is mostly related to the competition of individual trees in the stands, light interception, growth, and yield of trees. A total of 76 sample trees with a total number of 889 whorls and 3658 live branches were used to develop the outer crown profile model of the planted Pinus sylvestris var. mongolica trees in Heilongjiang Province, China. The power-exponential equation, modified Kozak equation, and simple polynomial equation were used and the model which showed the best performance was used as the basic model. The dummy variable approach was used to analyze the effect of stand age and stand density on the crown profile. Quantile regression for linear mixed-effects model, where the correlations between the series measurements on the same subject were considered, was used to model the outer crown profile. The results indicated that the power-exponential equation had the smallest error and was used as the basic model. Based on the dummy variable approach, stand age and stand density showed significant effects on the crown profile on the whole. Thus, they were directly included into the linear form of the power-exponential equation by a natural logarithm transformation to develop the quantile regression for the linear mixed-effects model. The 0.95th quantile regression model performed best in modeling the outer crown profile when compared to other quantiles. The prediction accuracy of the 0.95th quantile regression model by adding the random effects increased when compared to the quantile regression without random effect. The quantile regression for the linear mixed-effects model also showed an excellent performance in the largest crown radius prediction when compared to the quantile regression model. From suppressed trees to dominant trees, the crown radius increased, with tree size increasing for the same stand age and stand density increases. The crown radius of the suppressed trees from 21 to 40 year stands was the largest and the smallest was from older (>40 years) stands. The crown radius for both the intermediate and dominant trees from 21 to 40 year stands were similar and were larger than the younger (10–20 years) stands. The crown radius increased with tree size when the stand variables were constant. Furthermore, the crown radius increased with the increase of stand age, decreased with increasing stand density, and decreased with increased ratio of tree height to diameter at the breast height (HD) for trees with the same tree variables. Stand density had a weaker effect on the crown profile when compared to the HD. The growth rate of the crown radius of planted Pinus sylvestris var. mongolica trees increased with increasing stand age, and decreased with decreasing stand density. The growth rate of the crown radius decreased with increasing HD.