Superellipse Equation Describing the Geometries of Tree Rings.

Our previous study using 41 tree rings of one 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 discs collected from five sites in Denmark. The adjusted root-mean-square-error (RMSE) was calculated to assess the goodness of fit between observed and predicted tree-ring boundaries. The results showed that RMSE ranged between 0.0038 and 0.0591, with a mean value of 0.0141. This verified that the superellipse equation sufficiently describes the 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 was 0.95, 94% of tree rings had -values between 0.90 and 1.00, and only 67 tree rings had -values between 0.71 and 0.90. -value ranged from 1.62 to 2.81, with an average value of 2.04. 59% of the tree rings had -values between 1.90 and 2.10, and 62% showed -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 tends to be bilaterally symmetric and hyperellipse approached ellipse. Its variation was reflected more in inter-annual differences in - and -values.