Additivity of nonlinear tree crown width models: Aggregated and disaggregated model structures using nonlinear simultaneous equations

Yakai Lei, Liyong Fu, David L.R. Affleck, Andrew S. Nelson, Chenchen Shen, Mengxi Wang, Jingbiao Zheng, Qiaolin Ye, Guowei Yang

Research output: Contribution to journalArticlepeer-review

25 Scopus citations


A tree crown is a mass of foliage and significantly correlated to biomass and growth of other parts of a tree. Crown width (CW) is defined as an average of two crown diameters perpendicular to each other, obtained from measurements of four crown radii (crown components). The CW is often used as a covariate predictor in various forest models and therefore, its accurate estimate can be useful for forest management. Although various CW models have been developed so far, all of them lack a detailed investigation of the additivity properties of crown components and inherent correlations among the components and total CW. This study compared three alternative procedures of developing the systems of CW models: nonlinear simultaneous equations (NSE), disaggregated model structure with both one-step proportional weighting system (DMS&OSPWS), and two-step proportional weighting system, and aggregated model structure with one constraint on the total CW. Other two commonly used additivity methods: adjustment in proportion and ordinary least square with separating regression were also evaluated. The comparisons were based on the analyses of data obtained from 2207 trees of Mongolian oak (Quercus mongolica Fisch.) on 116 permanent sample plots located in northern China. Results showed that each method ensured a summation of crown components equal to twice as much as the total CW. The NSE better accounted for the inherent correlations among the crown components and total CW. Out of the model structures evaluated, DMS&OSPWS provided the best performance. This article emphasized more on the methodology and it was expected that the method could be applied by other researchers to develop similar systems of the CW models for other species as well.

Original languageEnglish
Pages (from-to)372-382
Number of pages11
JournalForest Ecology and Management
StatePublished - Nov 1 2018


  • Additivity property
  • Adjustment in proportion
  • Crown width
  • Nonlinear seemingly unrelated regression
  • Nonlinear simultaneous equations
  • Ordinary least squares with separating regression


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