Ellipse or superellipse for tree-ring geometries? evidence from six conifer species

Tree-ring shapes of the six studied coniferous species tend to be bilaterally symmetrical, and the superellipse equation is sufficient to describe the tree-ring boundaries and estimate the basal area increment.

In nature, under environmental pressures, such as wind, slope, water availability, etc., tree-ring shapes in most cases appear to be elliptical rather than circular. Compared with the ellipse equation, the superellipse equation includes an additional parameter that allows the generation of a larger range of geometries: hypoellipse, ellipse, and hyperellipse. The more complex Gielis equation can generate asymmetrical shapes. In the present study, we modeled the geometries of tree-rings for six coniferous species using the superellipse equation (i.e., the three-parameter model) and the more complex Gielis equation (i.e., the five-parameter model). The species-specific mean value of n approached 2 and the k-value was lower than 1, which confirmed that most tree-ring shapes of the studied coniferous species were closer to an ellipse rather than a circle. However, based on superellipse equation the n-value and k-value both showed an inter-annual fluctuation that ranged between 1.75–2.25 and 0.82–1.00, respectively. This suggests that most samples of tree-rings did not follow the typical ellipse equation, but the superellipse equation. Although the Gielis equation is slightly better in the goodness of fit than the superellipse equation, 86.67% of the percent errors (PEs) of RMSE_adj between these two equations were smaller than 5%, which means that the superellipse equation is better given the trade-off between the model structural complexity and goodness of fit. Most tree-ring shapes tend to be bilaterally symmetrical, and the three-parameter superellipse equation was verified to fit the tree-ring boundaries and estimate the inter-annual increments of tree-ring area well.