Evaluation of the box-counting method for assessing tree structures: A case study of European beech (Fagus sylvatica L.)

The box-counting method is sensitive to tree positions in the coordinate system, and based on this method, assuming self-similarity of beech tree models derived from mobile laser scanning was difficult. The repetitive branching architectures of trees lead us to think that trees exhibit self-similarity and fractal properties. Therefore, applying fractal analysis to assess tree structures may seem natural. This study aimed to evaluate the box-counting method (BCM), a simple and commonly used fractal analysis, for describing 3D biological trees, using a wide range of structural models of European beech trees (Fagus sylvatica L.) derived from mobile laser scanning. We specifically investigated the method’s sensitivity to the arbitrary placement of a tree within the coordinate system and the validity of the tree’s self-similarity based on the BCM. The BCM was sensitive to the tree position in the coordinate system, with observed minimum and maximum variations in the box-counting dimension (Db) values of 0.18 and 0.52, respectively, when translating the trees in the XYZ directions. The analysis of the local slopes of the BCM, which are the slopes between neighboring data point pairs, revealed a large variation of slope values across scales with a clear pattern, indicating that the structural patterns of the sampled trees were inconsistent and locally dependent. Thus, it is difficult to assume self-similarity of the beech tree models based on the BCM. Our results demonstrate the need to standardize the computation of the Db for single trees with respect to the coordinate system and caution in interpreting the Db. These findings contribute to a deeper understanding of the Db to assess tree structures and functions.