Superellipse Equation Describing the Geometries of Abies alba Tree Rings

Our previous study using 41 tree rings of one Abies alba Mill. disc indicated that the superellipse equation can accurately fit its tree-ring shape. This study further used the superellipse equation (xan+yβn=1 ) to model the geometries of 1090 tree rings of A. alba discs collected from five sites in Denmark. The adjusted root-mean-square-error (RMSE_adj) was calculated to assess the goodness of fit between observed and predicted tree-ring boundaries. The results showed that RMSE_adj ranged between 0.0038 and 0.0591, with a mean value of 0.0141. This verified that the superellipse equation sufficiently describes the A. alba tree-ring shape. In the polar coordinate system, the superellipse equation can be expressed as r=a(cosφn+sinφ/kn)-1/n. Where r and φ are the polar radius and polar angle, respectively. k=β/a, where a and β are the major and minor semi-axes of the superellipse. The mean value of k was 0.95, 94% of tree rings had k-values between 0.90 and 1.00, and only 67 tree rings had k-values between 0.71 and 0.90. n-value ranged from 1.62 to 2.81, with an average value of 2.04. 59% of the tree rings had n-values between 1.90 and 2.10, and 62% showed n-values greater than 2.0. This means that most tree rings are a hyperellipse approached to an ellipse. Sites with different soil moisture conditions influenced the size but not the shape of tree rings. This study verified that the tree-ring shape of A. alba tends to be bilaterally symmetric and hyperellipse approached ellipse. Its variation was reflected more in inter-annual differences in k- and n-values.