Introduction
Forests are the largest terrestrial ecosystem covering one third of the earth’s surface area (Roxburgh & Noble 2001) and they provide a range of services such as carbon uptake (Hardiman et al. 2011), productivity (Puettman et al. 2015), biodiversity (Fedrowitz et al. 2014), and resilience (Messier et al. 2013). Processes of growth and regeneration are closely related with these services also linking them with forest structure (von Gadow et al. 2012). The current forest structure is a result of tree and stand dynamics affected by the availability of resources such as light, nutrients, and water as well as by the competition of these resources. Both biotic (e.g. insects, pathogens) and abiotic (e.g. fire, wind, snow) disturbances as well as forest management and changing climate alter relationships between trees through changes in these growing conditions (i.e. availability of light, nutrients, and water) and therefore stand dynamics and forest structure.
Trees are interacting with each other and that affects their functioning and structure. Tomlinson (1983) has pointed out that development of trees and their structure can therefore enhance our understanding about forest structure. Thus, investigations on individual trees is important. Trees occupy three-dimensional space and tree architecture can be characterized based on growth dynamics and branching patterns (Tomlinson 1983). Tree structure, on the other hand, can be characterized by using morphological measures such as crown dimension (e.g. volume, surface area) and stem attributes (e.g. diameter at breast height (DBH), height, height of crown base) (Pretzsch 2014). The availability of 3D point clouds from terrestrial laser scanning (TLS) has provided an effective means for such measurements allowing TLS to be utilized in generating stem and crown attributes (Seidel et al. 2011, Liang et al. 2012, Bayer et al. 2013, Calders et al. 2013, Metz et al. 2013, Juchheim et al. 2017, Saarinen et al. 2017, Calders et al. 2018, Georgi et al. 2018, Saarinen et al. 2020). However, objective and quantitative measures for structural complexity of individual trees are needed to better understand relationship between forest structural diversity and ecosystem services such as biodiversity, productivity, and carbon uptake (Hardiman et al. 2011, Messier et al. 2013, Puettmann et al. 2015, Zenner 2015).
Fractal analysis (Mandelbrot 1977, Shenker 1994) can provide an approximation of natural forms and TLS has opened possibilities for applying fractal analysis for characterizing structural complexity of individual trees (Calders et al. 2020). Seidel (2018) presented an approach where fractal analysis of Minkowski-Bouligand dimension (or box-counting dimension, i.e. changes in number of boxes required covering an object when the boxes are made more defining) was applied in characterizing structural complexity of individual trees. Even before TLS existed, the so-called box dimension was used to characterize spatial patterns of foliage distribution with plastic flaps of different sizes to measure the presence of leaves (Osawa & Kurachi 2004). Seidel (2018) used boxes (or voxels) of different sizes to enclose all 3D points from individual trees obtained with TLS whereas Osawa & Kurachi (2004) used cylinders for estimating box dimension. Regardless of the geometric primitive, the box dimension is determined as a relationship between the number of primitives of varying size needed to enclose all 3D points of a tree and the inverse of the primitive size. The box dimension is scale-independent and can theoretically vary between one and three, one being a cylindrical, pole-like object and three corresponding solid objects such as cubes (Figure 1). Seidel et al. (2019a) assumed that maximum box dimension value for trees would be 2.72 that corresponds to the fractal object of a Menger sponge, which has infinite surface area with zero volume (Mandelbrot 1977, Pickover 2009)