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)