Modeling size-density trajectories for even-aged beech (Fagus silvatica L.) stands in France

We studied the size-density trajectories of pure even-aged unthinned beech stands in the ranges of 625–40,000 trees per hectare initial densities and of 12–33 years of age. A new piecewise polynomial function family was fitted to the trajectories, giving way to various applications. Initial number of stems per hectare (N0) and mean girth at breast height at the onset of mortality (Cg0) were parameters of the trajectory model, in addition to the parameters of the maximum size-density line. The two former parameters were tied by a linear relationship, which allowed the prediction of trajectories not considered in this study. Furthermore, the generic trajectory equation fitted the trajectories of thinned stands not used in the estimation of the parameters.


This paper models the size-density trajectories of pure even-aged beech stands, including the early development stage, which is not as well documented as are the later stages. The work reported in this paper concerns the development of a novel approach to size-density trajectories, considered as a mortality model to provide references to managers of beech forests. A 33-year-old beech spacing trial beginning at 12 years of age provided the opportunity to study the size-density trajectories of unthinned stands of this species. The beech data helped us to develop a new piecewise function to model these trajectories. The model we chose was a polynomial segment smoothly joining two linear functions. The fits of this model allowed us to estimate the parameters of the size-density trajectories of all stands, which were the quadratic mean girths at mortality onset and at maximum density. A linear relationship between these characteristics allowed us to reduce the number of parameters needed to fit the trajectories and made it possible to predict a stand trajectory from any initial density not observed in the experimental stands. A single-parameter function family could be used to fit the size-density trajectories of beech stands. The predicted trajectories have various applications in beech silviculture and growth simulators.

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Ningre F, Ottorini J-M, Le Goff N 2016. Modeling size-density trajectories for even-aged beech (Fagus silvatica L.) stands in France. Ann. For. Sci.: 1-12. 10.1007/s13595-016-0567-0.

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