Superellipse Equation Describing the Geometries of Abies alba Tree Rings - Weiwei Huang; Kehang Ma; Jiaxin Tan; Meixiang Wei; Yunjie Lu (Plants)
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 (<semantics><mfenced open="|" close="|" separators="|"><mstyle scriptlevel="0" displaystyle="true">xa</mstyle></mfenced>n+<mfenced open="|" close="|" separators="|"><mstyle scriptlevel="0" displaystyle="true">yβ</mstyle></mfenced>n=1</semantics> ) to model the geometries of 1090 tree rings of A. alba discs collected from five sites in Denmark. The adjusted root-mean-square-error (RMSEadj) was calculated to assess the goodness of fit between observed and predicted tree-ring boundaries. The results showed that RMSEadj 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 <semantics>r=a(<mfenced open="|" close="|" separators="|">cosφ</mfenced>n+<mfenced open="|" close="|" separators="|">sinφ/k</mfenced>n)−1/n</semantics>. Where <semantics>r</semantics> and <semantics>φ</semantics> are the polar radius and polar angle, respectively. <semantics>k=β/a</semantics>, where <semantics>a</semantics> and <semantics>β</semantics> 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.