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\begin{document}

\title{Understanding 3D structural complexity of individual Scots pine trees
with different management history}



\author[1]{Ninni Saarinen}%
\author[2]{Kim Calders}%
\author[3]{Ville Kankare}%
\author[4]{Tuomas Yrttimaa}%
\author[4]{Samuli Junttila}%
\author[5]{Saija Huuskonen}%
\author[5]{Jari Hynynen}%
\author[6]{Hans Verbeeck}%
\affil[1]{University of Helsinki Faculty of Agriculture and Forestry}%
\affil[2]{Ghent University Faculty of Bioscience Engineering}%
\affil[3]{University of Eastern Finland}%
\affil[4]{University of Eastern Finland Faculty of Science and Forestry}%
\affil[5]{Natural Resources Institute Finland}%
\affil[6]{Universiteit Gent}%


\vspace{-1em}



  
  \date{May 21, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
Tree functional traits together with processes such as forest
regeneration, growth, and mortality affect forest and tree structure.
Forest management inherently impacts these processes. Moreover, forest
structure, biodiversity, resilience, and carbon uptake can be sustained
and enhanced with forest management activities. To assess structural
complexity of individual trees, comprehensive and quantitative measures
are needed, and they are often lacking for current forest management
practices. Here, we utilized 3D information from individual Scots pine
(Pinus sylvestris L.) trees obtained with terrestrial laser scanning
(TLS) to first, assess effects of forest management on structural
complexity of individual trees, and second, understand relationship
between several tree attributes and structural complexity. We studied
structural complexity of individual trees represented by a single scale
independent metric called ``box dimension''. This study aimed at
identifying drivers affecting structural complexity of individual Scots
pine trees in boreal forest conditions. The results showed that thinning
increased structural complexity of individual Scots pine trees.
Furthermore, we found a relationship between structural complexity and
stem and crown size and shape as well as tree growth. Thus, it can be
concluded that forest management affected structural complexity of
individual Scots pine trees in managed boreal forests, and stem, crown,
and growth attributes were identified as drivers of it.%
\end{abstract}%



\sloppy


\textbf{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)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image1/image1}

\end{center}
\end{figure}

Figure 1. Examples of objects with box dimension ranging from one
(cylindrical pole) to three (solid cube) in between two real-life trees
and a Menger sponge (box dimension = 2.72).

Box dimension is a relatively new measure for assessing structural
complexity of trees and forests in relation to TLS. Seidel et al.
(2019b) studied the relationship between structural complexity (i.e. box
dimension) and horizontal and vertical architectural characteristics
(i.e. tree height and volume, crown radius and surface area, branching
angles) of deciduous trees (\emph{Fagus sylvatica, Fraxinus excelsior,
Acer pseudoplatanus, Carpinus betulus} ) of varying size. They concluded
that structural complexity was related to crown radius and surface area
of deciduous trees. Dorji et al. (2019) studied how competition affects
structural complexity of European beech (\emph{Fagus sylvatica} L.)
trees and concluded that their crowns were influenced by competition, as
measured through box dimension. Seidel et al. (2019a), on the other
hand, reported decreasing structural complexity when competition (i.e.
light availability) increased for deciduous trees. Previous work (Seidel
et al. 2018, 2019a, b) demonstrated the potential of box dimension as a
meaningful measure for structural complexity of individual trees.
However, how this measure can be used to quantify forest structure of
conifers and how it can expand our understanding about effects of
anthropogenic activities (e.g. forest management) on tree structure is
largely unexplored.

Although forest management affects growing conditions of trees as well
as their size and shape (M\selectlanguage{ngerman}äkinen \& Isomäki 2004, Saarinen et al. 2020),
it is unclear how forest management affects structural complexity of
conifers. This study aims at identifying relationships between a variety
of attributes (e.g. characterizing stem, crown, and competition) and
structural complexity of individual Scots pine (\emph{Pinus
sylvestris}L.) trees in even-aged and single-layered managed boreal
forest conditions. We aim to understand how structural complexity is
driven by forest management and underlying structural attributes. We
hypothesize that thinning intensity affects structural complexity of
Scots pine trees (H1). It is also hypothesized that horizontal and
vertical measures (e.g. stem and crown dimensions) are related to
structural complexity of Scot pine trees (H2). Finally, we hypothesize
(H3) that there is a relationship between structural complexity and i)
crown dimensions, ii) architectural benefit-to-cost ratio (i.e.
surface-to-volume ratio), iii) tree growth (i.e. DBH, height, volume,
and \selectlanguage{greek}Δ\selectlanguage{english}height/DBH), and iv) the availability of light (i.e. competition).
In other words, structural complexity of individual Scots pine trees can
be explained by these attributes.

\textbf{Materials and methods}

\emph{Study site and data acquisition}

The study area consists of three study sites dominated by Scots pine
(Palom\selectlanguage{ngerman}äki, Pollari, and Vesijako) (Figure 2), established and maintained
by Natural Resources Institute Finland (Luke). All study sites are
located in southern boreal forest zone and characterized as mesic heath
forest (i.e., Myrtillus forest site type according to Cajander (1913)).
Nine rectangular sample plots (size from 900 and 1200
m\textsuperscript{2}) were placed at each study site resulting in a
total of 27 sample plots. At the time of the establishment (Palomäki in
2005, Pollari and Vesijako in 2006), the stand age was 50, 45, and 59
years for Palomäki, Pollari, and Vesijako, respectively. The first
thinning (removal \textasciitilde{}30\% of stems) had been carried out
for all study sites in the early 1990s.

The experimental design of the study sites includes two varying levels
of thinning intensity (i.e. moderate and intensive) and one plot at each
study site remained as a control plot where no thinning has been carried
out since the establishment of the sites. The remaining relative stand
basal area after moderate thinning was \textasciitilde{}68\% of the
stocking before thinning and intensive thinning reduced the stocking
levels down to 34\%. Suppressed and co-dominant as well as unsound and
damaged (e.g. crooked, forked) trees were removed (i.e. thinning from
below) in both thinning intensities. Other thinning treatments were also
carried out but here we concentrated on 3 plots with moderate, 3 plots
with intensive, and 3 plots with no treatment since establishment.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}

\end{center}
\end{figure}\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}

\end{center}
\end{figure}

Figure 2. Location of the three study sites namely Palom\selectlanguage{ngerman}äki, Pollari,
and Vesijako and vegetation zones in Finland (left) and study sites on
top of satellite imagery © 2020 TerraMetrics.

The plots were measured and thinning treatments performed at the time of
the establishment (in 2005 at Palomäki and in 2006 at Pollari and
Vesijako). The most recent field measurements were carried out in
October 2018 and in April 2019. Tree species, crown layer, DBH from two
perpendicular directions, and health status were recorded from each tree
within a plot (i.e. tally trees). At the time of the latest
measurements, the proportion of Norway spruce and deciduous trees
(i.e.\emph{Betula} sp and \emph{Alnus} sp) from the total stem volume of
all trees within the 9 sample plots was 3.02\% and 0.07\%, respectively.
Approximately half of the trees (n=318) were selected as sample trees
from which tree height, crown base height, and height of the lowest dead
branch were also measured. Height of the tally trees was estimated using
an allometric model calibrated for each sample plot with the information
from the sample trees. Stem volume of individual trees was produced by
using the existing nationwide, species-specific volume equations with
DBH and height as predictors (Laasasenaho 1982) to obtain corresponding
information from different field measurements. Plot-level attributes
before and after thinning treatments are presented in Table 1, and the
development of tree-level attributes for each thinning treatment can be
found in Table 2.

Table 1. Mean stand characteristics by treatments before and after
thinning and thinning removal. N = stem number per hectare, G = basal
area, D\textsubscript{w} = mean diameter weighted by basal area,
H\textsubscript{w} = mean height weighted by basal area,
H\textsubscript{100} = dominant height, and V = volume.\selectlanguage{english}
\begin{longtable}[]{@{}llllllll@{}}
\toprule
Before thinning (2005-2006) & Before thinning (2005-2006) & Before
thinning (2005-2006) & Before thinning (2005-2006) & After thinning
(2005-2006) & After thinning (2005-2006) & After thinning (2005-2006)
&\tabularnewline
\midrule
\endhead
& \textbf{Moderate} & \textbf{Intensive} & \textbf{No treatment} &
\textbf{Moderate} & \textbf{Intensive} & \textbf{No treatment} &
\textbf{No treatment}\tabularnewline
N/ha & 1269 & 1244 & 1337 & 716 & 289 & 1337 & 1337\tabularnewline
G (m\textsuperscript{2}/ha) & 26.5 & 26.5 & 27.7 & 18.1 & 8.7 & 27.7 &
27.7\tabularnewline
D\textsubscript{w} (cm) & 17.5 & 18.0 & 17.8 & 18.7 & 20.4 & 17.8 &
17.8\tabularnewline
H\textsubscript{w} (m) & 16.1 & 16.3 & 16.1 & 16.5 & 16.9 & 16.1 &
16.1\tabularnewline
H\textsubscript{100} (m) & 17.3 & 17.7 & 17.5 & 17.3 & 17.5 & 17.5 &
17.5\tabularnewline
V (m\textsuperscript{3}/ha) & 213.4 & 215.7 & 224.0 & 148.3 & 72.9 &
224.0 & 224.0\tabularnewline
\bottomrule
\end{longtable}

Table 2. Mean tree-level attributes with its standard deviation (with \selectlanguage{ngerman}±)
and range in square brackets for each treatment at the year of the
establishment (First measurement) and the last measurement. DBH =
diameter at breast height.\selectlanguage{english}
\begin{longtable}[]{@{}lllllll@{}}
\toprule
\begin{minipage}[b]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut
First measurement (2005-2006)\strut
\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut
First measurement (2005-2006)\strut
\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut
First measurement (2005-2006)\strut
\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut
Last measurement (2018-2019)\strut
\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut
Last measurement (2018-2019)\strut
\end{minipage} & \begin{minipage}[b]{0.12\columnwidth}\raggedright\strut
Last measurement (2018-2019)\strut
\end{minipage}\tabularnewline
\midrule
\endhead
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\textbf{Moderate}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\textbf{Intensive}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\textbf{No treatment}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\textbf{Moderate}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\textbf{Intensive}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
\textbf{No treatment}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
DBH (cm)\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
17.6\selectlanguage{ngerman}±3.3 {[}10.3; 28.2{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
19.3\selectlanguage{ngerman}±3.4 {[}11.7; 29.8{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
15.4\selectlanguage{ngerman}±4.6 {[}5.8; 30.7{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
22.2\selectlanguage{ngerman}±3.7 {[}13.4; 35.3{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
26.4\selectlanguage{ngerman}±3.9 {[}17.9; 36.4{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
18.7\selectlanguage{ngerman}±5.0 {[}6.5; 34.4{]}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
Height (m)\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
15.9\selectlanguage{ngerman}±1.9 {[}11.9; 19.9{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
16.5\selectlanguage{ngerman}±1.8 {[}13.0; 20.5{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
14.7\selectlanguage{ngerman}±2.6 {[}9..7; 23.0{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
21.2\selectlanguage{ngerman}±2.1 {[}14.7; 25.9{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
21.2\selectlanguage{ngerman}±1.7 {[}18.2; 25.3{]}\strut
\end{minipage} & \begin{minipage}[t]{0.12\columnwidth}\raggedright\strut
20.0\selectlanguage{ngerman}±3.0 {[}13.2; 30.3{]}\strut
\end{minipage}\tabularnewline
\begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
Volume (dm\textsuperscript{3})\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
202.7\selectlanguage{ngerman}±89.3 {[}50.9; 549.4{]}\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
249.1\selectlanguage{ngerman}±107.0 {[}75.6; 623.0{]}\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
160.5\selectlanguage{ngerman}±119.7 {[}13.0; 783.6{]}\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
408.3\selectlanguage{ngerman}±160.3 {[}116.6; 1050.8{]}\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
563.8\selectlanguage{ngerman}±202.5 {[}231.9; 1146.2{]}\strut
\end{minipage} & \begin{minipage}[t]{0.14\columnwidth}\raggedright\strut
299.4\selectlanguage{ngerman}±190.8 {[}20.9; 1266.4{]}\strut
\end{minipage}\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{ngerman}

TLS data acquisition was carried out with a Trimble TX5 3D phase-shift
laser scanner (Trimble Navigation Limited, USA) operating at a 1550 nm
wavelength and measuring 976,000 points per second, delivering a
hemispherical (300° vertical x 360° horizontal) point cloud with an
angular resolution of 0.009° in both vertical and horizontal direction
with a maximum range of 120 m (resulting a point distance approximately
6.3 mm at 10-m distance) and beam divergence of 0.011°. All three study
sites were scanned between September and October 2018 using a multi-scan
approach to minimize occlusion. Eight scans were conducted at each
sample plot with two scans on two sides of a plot center and six
auxiliary scans closer to the plot borders (see Figure 1 in Saarinen et
al. 2020). Artificial targets (i.e. white spheres with a diameter of 198
mm) were placed around each sample plot to be used as reference objects
for registering the eight scans into a single, aligned coordinate system
with a FARO Scene software (version 2018). The registration resulted in
a mean distance error of 2.9±1.2 mm, mean horizontal error was 1.3±0.4
mm, and mean vertical error 2.3±1.2 mm. LAStools software (Isenburg
2019) was used to remove topography from the point clouds by applying a
point cloud normalization workflow presented by Ritter et al. (2017).

\textbf{Methods}

\emph{TLS point cloud classification into stem and non-stem}

First, plot-level TLS point clouds were segmented to identify points
from individual trees. Local maxima from canopy height models (CHMs)
with a 20-cm resolution were identified using the Variable Window Filter
approach (Popescu \& Wynne 2004) and the Marker-Controlled Watershed
Segmentation (Meyer \& Beucher 1990) was applied to delineate crown
segments. A point-in-polygon approach was applied for identifying all
points belonging to each crown segment. To identify points that
originated from stem and crown within each crown segment, a point cloud
classification procedure by Yrttimaa et al (2020) was used. The
classification of stem and non-stem points assumed that stem points have
more planar, vertical, and cylindrical characteristics compared to
non-stem points representing branches and foliage (Liang et al. 2012,
Yrttimaa et al. 2020). The method by Yrttimaa et al. (2019, 2020) is an
iterative procedure beginning from the base of a tree and proceeding
towards treetop. More detailed description of the point cloud
classification workflow can be found in Yrttimaa et al. (2019, 2020).
The result of this step was 3D point clouds for each individual tree
(n=741) within the 9 sample plots (Figure 3A).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}

\end{center}
\end{figure}

Figure 3. Crown-segmented point clouds of individual Scots pine trees (A
bottom) and an example of classified point clouds representing a Scots
pine tree (A top center) with the fitted 3D convex hull enveloping the
crown points, viewed from the top (A top left) and side (A top right).
The definition for the box dimension for the same Scots pine (B), the
slope of the fitted straight line (1.90) equals the box dimension
whereas the intercept (-0.27) is a measure of tree size and coefficient
of determination (R\textsuperscript{2}=1.0) self-similarity.

\emph{Attributes for structural complexity, crown dimensions,
benefit-to-cost ratio, growth, and light availability}

Box dimension was used for assessing structural complexity of the
individual trees. Box dimension is a structural measure derived from
individual tree TLS point clouds. First, one box including all TLS
points of a single tree was fitted (i.e. initial box) in which the edge
length of the box was tree height and then boxes of different sizes
(i.e. tree height/2, tree height/4, tree height/8, tree height/16, tree
height/32, tree height/64, tree height/128) were fitted to point clouds
of each tree and the number of fitted boxes of each size was saved.
Finally, the box dimension for each tree was defined as a slope between
natural logarithm of 1/(box edge length of certain size/edge length of
initial box) and natural logarithm of number of boxes including boxes of
certain size (Figure 3B). Box dimension can theoretically vary between
one and three, one representing pole-like objects and three solid
objects such as a cube

Following examples by Seidel et al (2019a, b), the relationship between
box dimension and attributes characterizing stem and crown size as well
as benefit-to-cost ratio, growth, and light availability were assessed.
Stem attributes included DBH, tree height, and stem volume whereas crown
attributes included crown radius, crown projection area, and crown
volume. Tree height was obtained using the height of the highest TLS
point of each tree whereas DBH was defined from taper curve obtained
with a combination of circle fitting to original stem points and fitting
a cubic spline (see Yrttimaa et al. 2019, Saarinen et al. 2020). Stem
volume, on the other hand, was defined by considering the stem as a
sequence 10 cm vertical cylinders and summing up the volumes of the
cylinders using the estimated taper curve. Crown attributes were
generated from TLS points originating from branches and foliage (i.e.
crown points). A 2D convex hull was fitted to envelope the crown points
of each tree of which crown projection area was derived whereas crown
volume was calculated from a 3D convex hull. Crown width, on the other
hand, was defined as the distance between the two most outer points in
xy-space.

Benefit-to-cost ratio was defined as a ratio between crown surface area
and stem volume (i.e. surface-to-volume ratio), which were used as
proxies for the photosynthetically active surface and building costs of
a tree, respectively. The crown surface area was calculated from a 3D
convex hull fitted to crown points of each tree (Figure 3A).

As TLS data were only acquired once from the study sites, growth of DBH,
tree height, stem volume, and \selectlanguage{greek}Δ\selectlanguage{english}H/DBH was calculated using field
inventory measurements conducted in 2005-2006 and 2018-2019 for all live
trees that were in the sample plots during the last field measurements.

Light availability of tree crowns was assumed to be related to the level
of competition each tree is facing and the Hegyi index was used as a
measure for competition and thus, a proxy for the light availability
(Seidel et al. 2019a). The Hegyi's competition index was calculated for
each tree as:

\(\text{Hegy}i^{{}^{\prime}}s\ competition\ index=\ \sum_{j=1}^{n}\frac{\frac{\text{DBH}_{i}}{\text{DBH}_{j}}}{\text{dist}_{\text{ij}}}\), (1)

where \emph{i} is the subject tree for which competition index is
calculated, \emph{j} is a competitor, \(\text{dist}_{\text{ij}}\) is the distance
between the subject tree \emph{i} and the competitor \emph{j} , and
\emph{n} is the number of competitors within 5-m radius around the
subject tree \emph{i} . The TLS-based DBH of subject and competitor
trees was used in calculating Hegyi's competition index and the RMSE and
bias are 0.7 cm (3.4\%) and -0.1 (-0.6\%), respectively (Yrttimaa et al.
2020).

\emph{Statistical analyses}

Due to the data structure (i.e. several sample plots in each study
site), a nested two-level linear mixed-effects model (Equation 2) was
fitted using Restricted Maximum Likelihood included in package nlme
(Pinheiro et al. 2020) of the R-software to assess the effects of
thinning treatment on box dimension.

\(y_{\text{ij}}=\ {\beta_{1}\text{Moderate\ below}}_{i}+{\beta_{2}\text{Moderate\ above}}_{i}+{\beta_{3}\text{Moderate\ systematic}}_{i}+{\beta_{4}\text{Intensive\ below}}_{i}+{\beta_{5}\text{Intensive\ above}}_{i}+{\beta_{6}\text{Intensive\ systematic}}_{i}+{\beta_{7}\text{No\ treatment}}_{i}+a_{i}+c_{\text{ij}}+\epsilon_{\text{ij}},\)(2)

where \(y_{\text{ij}}\) is box dimension, \(\beta_{1},\ldots\beta_{7}\)are fixed
parameters, i, i = 1, \ldots{}, M, refers to study site, j, j = 1,
\ldots{}, \(n_{i}\), to a plot, \(a_{i}\) and
\(c_{\text{ij}}\) are normally distributed random effects for sample
plot \emph{j} and for sample plot \emph{j} within study site \emph{i} ,
respectively, with mean zero and unknown, unrestricted
variance-covariance matrix, and\(\epsilon_{\text{ij}}\) is a residual error
with a mean zero and unknown variance. The random effects are
independent across study sites and sample plots as well as residual
errors are independent across trees. The effects of a study site and a
sample plot within the study sites on box dimension, crown and stem
attributes, surface-to-crown ratio, growth, as well as light competition
were assessed through their variances.

The analysis of variance utilizing the results from the nested two-level
linear mixed-effects model was applied in testing the statistically
significant difference in the box dimension affected by the thinning
treatments, the study sites as well as the plots within the study sites.
Furthermore, to reveal the possible statistically significant difference
in the box dimension between a thinning treatment against other
treatments, Tukey's honest significance test was applied.

To assess the relationship between structural complexity and stem and
crown dimensions, benefit-to-cost ratio, as well as growth and light
availability, similar approach was applied but box dimension was added
as a continuous predictor variable into Equation 2. Then the response
variable was a single stem and crown attribute, benefit-to-cost ratio,
and growth attributes (i.e. DBH, height, stem volume, and \selectlanguage{greek}Δ\selectlanguage{english}H/DBH) at a
time.

The analysis of variance was applied to investigate the significance of
the relationship between box dimension and thinning treatment, whereas
Tukey's honest significance test was used for revealing difference in
architectural attributes (stem and crown dimensions,
benefit-to-cost-ratio, growth, and light availability) between thinning
intensity.

Finally, Pearson's correlation coefficient and coefficient of
determination (R\textsuperscript{2}) were calculated between box
dimension and stem and crown attributes as well as benefit-to-cost
ratio, growth attributes, and competition index for each thinning
treatment to assess their relationships.

\textbf{Results}

The box dimension was 1.5\selectlanguage{ngerman}±0.1 and 1.6±0.1 for moderate and intensive
thinnings, respectively, whereas for control plots it was 1.4±0.1,
indicating increasing structural complexity of individual trees as the
thinning intensity increased. The linear mixed-effects model and the
analysis of variance revealed statistically significant difference
(p\textless{}0.01) in box dimension between thinning treatments (Figure
4). Trees with small differences in box dimension can look very
different as depicted in Figure 5.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}

\end{center}
\end{figure}

Figure 4. Variation in box dimension between moderate and intensive
thinning as well as without thinning (no treatment).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}

\end{center}
\end{figure}

Figure 5. Terrestrial laser scanning point clouds from example trees
with varying structural complexity (i.e. box dimension).

When assessing drivers of structural complexity, all stem, crown, and
growth attributes were found significant (p\textless{}0.05) in the
nested two-level linear mixed-effects models (Table 3). Benefit-to-cost
ratio and light availability, on the other hand, were not. Intensive
thinning increased tree height, benefit-to-cost ratio, height growth,
and light availability significantly (p\textless{}0.05), whereas
benefit-to-costa ratio, height growth, light availability was
significantly (p\textless{}0.05) smaller in the linear mixed-effects
models (Table 3). However, coefficient of determination was
\textless{}0.2 for all other architectural attributes (Figures S1-S4)
except for crown dimensions where it was 0.5 between box dimension and
crown projection area and crown volume (Figure S5).

Thinning treatment was statistically significant (p\textless{}0.05) in
mixed-effects models where height and volume growth, benefit-to-cost
ratio, light availability, and each stem attribute was included as
predictor variable at a time. Tukey's honest significance test revealed
that there was a statistical difference (p\textless{}0.05) between
moderate and intensive thinning in all stem attributes, benefit-to-cost
ratio, height and volume growth, and light availability. Thinning,
either moderate or intensive, resulted in significant difference
(p\textless{}0.05) in all stem attributes, crown width, benefit-to-cost
ratio, all growth attributes, and light availability when compared to
trees without a thinning treatment.

Table 3. Results of the nested two-level linear mixed-effects models
with box dimension as dependent variable and thinning treatment together
with stem and crown attributes, benefit-to-cost ratio, growth
attributes, and light availability as independent variables. * denotes
statistical significance of p-value\textless{}0.05.\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
& Attribute & Model parameter & Estimate & SE & p-value\tabularnewline
\midrule
\endhead
Stem attributes & \textbf{DBH (cm)} & DBH & 0.001 & 0.001 &
0.000*\tabularnewline
& & Moderate & 1.312 & 0.031 & 0.000*\tabularnewline
& & Intensive & 0.087 & 0.036 & 0.074\tabularnewline
& & No treatment & -0.055 & 0.035 & 0.191\tabularnewline
& \textbf{Height (m)} & Height & 0.001 & 0.002 & 0.000*\tabularnewline
& & Moderate & 1.369 & 0.043 & 0.000*\tabularnewline
& & Intensive & 0.124 & 0.035 & 0.025*\tabularnewline
& & No treatment & -0.082 & 0.034 & 0.074\tabularnewline
& \textbf{Stem volume (dm\textsuperscript{3})} & Stem volume & 0.000 &
0.000 & 0.000*\tabularnewline
& & Moderate & 1.443 & 0.028 & 0.000*\tabularnewline
& & Intensive & 0.096 & 0.038 & 0.064\tabularnewline
& & No treatment & -0.066 & 0.037 & 0.148\tabularnewline
Crown attributes & \textbf{Crown width (m)} & Crown width & 0.065 &
0.004 & 0.000*\tabularnewline
& & Moderate & 1.232 & 0.025 & 0.000*\tabularnewline
& & Intensive & 0.064 & 0.027 & 0.078\tabularnewline
& & No treatment & -0.032 & 0.026 & 0.282\tabularnewline
& \textbf{Crown projection area (CPA) (m\textsuperscript{2})} & CPA &
0.014 & 0.001 & 0.000*\tabularnewline
& & Moderate & 1.356 & 0.022 & 0.000*\tabularnewline
& & Intensive & 0.023 & 0.024 & 0.395\tabularnewline
& & No treatment & -0.018 & 0.022 & 0.469\tabularnewline
& \textbf{Crown volume (m\textsuperscript{3})} & Crown volume & 0.001 &
0.000 & 0.000*\tabularnewline
& & Moderate & 1.364 & 0.027 & 0.000*\tabularnewline
& & Intensive & 0.038 & 0.301 & 0.282\tabularnewline
& & No treatment & -0.025 & 0.030 & 0.458\tabularnewline
Benefit-to-cost ratio & \textbf{Surface-to-volume ratio (StV)} & StV &
0.000 & 0.000 & 0.063\tabularnewline
& & Moderate & 1.154 & 0.022 & 0.000*\tabularnewline
& & Intensive & 0.125 & 0.031 & 0.015*\tabularnewline
& & No treatment & -0.092 & 0.029 & 0.036*\tabularnewline
Growth attributes & \textbf{DBH growth (cm)} & DBH growth & 0.023 &
0.003 & 0.000*\tabularnewline
& & Moderate & 1.149 & 0.025 & 0.000*\tabularnewline
& & Intensive & 0.065 & 0.033 & 0.118\tabularnewline
& & No treatment & -0.041 & 0.0032 & 0.259\tabularnewline
& \textbf{Height growth (m)} & Height growth & 0.016 & 0.004 &
0.000*\tabularnewline
& & Moderate & 1.442 & 0.031 & 0.000*\tabularnewline
& & Intensive & 0.131 & 0.029 & 0.011*\tabularnewline
& & No treatment & -0.080 & 0.0282 & 0.046*\tabularnewline
& \textbf{Volume growth (dm\textsuperscript{3})} & Volume growth & 0.000
& 0.000 & 0.000*\tabularnewline
& & Moderate & 1.452 & 0.024 & 0.000*\tabularnewline
& & Intensive & 0.085 & 0.033 & 0.062\tabularnewline
& & No treatment & -0.056 & 0.032 & 0.155\tabularnewline
& \selectlanguage{greek}\textbf{ΔH/DBH}\selectlanguage{english} & \selectlanguage{greek}Δ\selectlanguage{english}H/DBH & 0.526 & 0.081 & 0.000*\tabularnewline
& & Moderate & 1.025 & 0.071 & 0.000*\tabularnewline
& & Intensive & 0.065 & 0.033 & 0.125\tabularnewline
& & No treatment & -0.041 & 0.032 & 0.271\tabularnewline
Light availability & \textbf{Competition index (CI)} & CI & 0.004 &
0.004 & 0.279\tabularnewline
& & Moderate & 1.518 & 0.022 & 0.000*\tabularnewline
& & Intensive & 0.120 & 0.033 & 0.022*\tabularnewline
& & No treatment & -0.095 & 0.031 & 0.040*\tabularnewline
\bottomrule
\end{longtable}

Pearson's correlation coefficient was \textgreater{}0.5 between box
dimension and crown projection area and crown volume with moderate no
thinning (Table 4). However, the correlation was significant
(p\textless{}0.001) between box dimension and all crow attributes with
all thinning treatments. However, the correlation between box dimension
and stem attributes, benefit-to-cost ratio, growth, as well as light
availability was mostly \textless{}0.5 indicating a weak relationship
between them. Nevertheless, the Person's correlation coefficients were
found significant for stem attributes in moderate thinning and for
growth attributes in control plots.

Table 4. Pearson's correlation coefficients between box dimension and
stem and crown attributes, benefit-to-cost ratio, growth attributes, and
light availability grouped by thinning treatment. Correlation
coefficients \textgreater{}0.50 are bolded and * denotes statistical
significance (p\textless{}0.001).\selectlanguage{english}
\begin{longtable}[]{@{}llllll@{}}
\toprule
& & Moderate & Intensive & No treatment &\tabularnewline
\midrule
\endhead
& & \textbf{Box dimension} & \textbf{Box dimension} & \textbf{Box
dimension} & \textbf{Box dimension}\tabularnewline
Stem attributes & \textbf{DBH (cm)} & 0.32* & 0.27 & 0.33*
&\tabularnewline
& \textbf{Height (m)} & 0.28* & -0.08 & -0.01 &\tabularnewline
& \textbf{Stem volume (dm\textsuperscript{3})} & 0.24* & 0.17 & 0.24
&\tabularnewline
Crown attributes & \textbf{Crown width (m)} & \textbf{0.55*} & 0.36* &
\textbf{0.56*} &\tabularnewline
& \textbf{Crown projection area (m\textsuperscript{2})} & \textbf{0.69*}
& 0.43* & \textbf{0.62*} &\tabularnewline
& \textbf{Crown volume (m\textsuperscript{3})} & \textbf{0.67*} & 0.37*
& \textbf{0.57*} &\tabularnewline
Benefit-to-cost ratio & \textbf{Surface-to-volume ratio} & 0.37* & 0.00
& 0.06 &\tabularnewline
Growth attributes & \textbf{DBH growth (cm)} & 0.09 & 0.34* & 0.43*
&\tabularnewline
& \textbf{Height growth (m)} & -0.03 & -0.09 & 0.23* &\tabularnewline
& \textbf{Volume growth (dm\textsuperscript{3})} & 0.26* & 0.23 & 0.25*
&\tabularnewline
& \selectlanguage{greek}\textbf{ΔH/DBH}\selectlanguage{english} & 0.07 & 0.34* & 0.33* &\tabularnewline
Light availability & \textbf{Competition index} & -0.25* & 0.06 & 0.08
&\tabularnewline
\bottomrule
\end{longtable}

\textbf{Discussion}

Thinning increased the structural complexity of individual Scots pine
trees confirming our hypothesis H1. Crown projection area showed a
positive relationship (mean correlation coefficient \textgreater{}0.6)
with structural complexity whereas tree height did not, leading to
partially accepting the hypothesis about associations between structural
complexity and horizontal and vertical measures of Scots pine trees
(H2). Finally, the hypothesis H3 is also partly accepted as there was
practically no relationship between structural complexity and
benefit-to-cost ratio, tree growth, or the light availability but crown
and growth attributes were, however, significant when estimating
structural complexity. Forest management, and thinning especially,
affected structural complexity of individual Scot pine trees. Stem,
crown, and growth variables were found significant predictors for it
indicating them as drivers for structural complexity.

Crown dimensions affected structural complexity more than stem
attributes (i.e. DBH, tree height, and volume) which is similar findings
to Seidel et al. (2019b) who studied the relationship between structural
complexity and stem and crown attributes of four deciduous species
(i.e.\emph{Fagus sylvatica} L., \emph{Fraxinus excelsior} L., \emph{Acer
pseudoplatanus} L., and \emph{Carpinus betulus} L.). Thinning affected
stem or growth attributes, benefit-to-cost ratio, and light availability
of Scots pine trees, as either moderate or intensive thinning resulted
in differing values for these attributes compared to Scots pine trees
without a thinning treatment (i.e. control plots).

There was no relationship (R\textsuperscript{2}\textless{}0.2 and
correlation coefficient \textless{}0.1, not significant) between
structural complexity and benefit-to-cost ratio indicating that it does
not affect structural complexity of Scots pine trees. This is
contradictory to the findings by Seidel et al. (2019a) who found
R\textsuperscript{2} of at least 0.25 between structural complexity and
cost-to-benefit ratio for 76 deciduous trees (i.e. 46 \emph{Fagus
sylvatica} L., 25 \emph{Fraxinus excelsior} L., and 5 \emph{Acer
pseudoplatanus} L.). Their study site was a mixed, unmanaged deciduous
forest and there were \textasciitilde{}150 trees/ha whereas in our study
the tree density per ha was at least twice of that. Thus, it can be
assumed that there was more space for the deciduous trees to grow.
Additionally, tree form in general is different between conifers and
deciduous trees due to the difference in their shoot growth, in other
words the degree of apical dominance differs being strong for conifers
and weaker for deciduous trees (Kozlowski 1964). This can produce
differences in structural complexity and its relationship to
benefit-to-cost ratio.

Light availability (measured through competition) did not affect
structural complexity of Scots pine trees and there was no relationship
between them. However, larger variation in light availability was found
in plots without thinning treatments, which is expected as it has shown
in previous studies that thinning decreases competition (M\selectlanguage{ngerman}äkinen \&
Isomäki 2004, Juchheim et al. 2017, Jacobs et al. 2020). Seidel et al.
(2019a) reported decreasing structural complexity when competition
increased but their study only included 93 read oak (\emph{Quercus
rubra} L.) trees from ten different sites compared to our
\textasciitilde{}740 Scot pine trees. Similar to the results of
benefit-to-cost ratio discussed above, deciduous trees have a different
branching pattern compared to conifers. Future research should focus on
the response of structural complexity to light availability of different
species with enough trees to clearly understand their relationship.

Ehbrecht et al. (2017) studied stand structural complexity in mixed
stands with even-aged and uneven-aged forest management. They discovered
that stand structural complexity differed between tree species and
even-aged coniferous stands had less complex structure compared
to\emph{Fagus sylvatica} (L.) stands. We only studied one tree species,
namely Scots pine, but found that structural complexity varied between
trees with different management history, although all the plots were
even-aged and single layer that could have decreased their structural
complexity.

Forest structural complexity has been widely studied (Camarretta et al.
2020, McElhinny et al. 2005, Ishii et al. 2004) but there is less
research on tree structural complexity, which was the focus of this
study. TLS can provide traditional tree attributes (i.e. DBH, volume,
crown dimensions) for characterizing tree structure but its potential
can also be expanded for new attributes. Here, we only utilized box
dimension to measure structural complexity, which can be considered as a
weakness of the study as we cannot compare these results to other
measures. However, the box dimension was chosen because it was easy to
calculate, it has shown its potential in characterizing individual tree
structure (Seidel et al. 2019a, b) , and because no other measures for
individual tree structural complexity was found in the literature. As it
was shown, there was a relationship between box dimension and stem,
crown, and growth attributes indicating that these attributes can
explain structural complexity if individual trees. None of them,
however, considers the entire 3D tree structure to which the box
dimension brings added value. The strength of the study is that the
study design allowed us to compare forest management practices without
confounding factors that could have been a challenge in uneven-aged or
mixed-species stands.

Forest management in general, and thinning in particular, controls the
between-tree competition especially by removing part of shadowing canopy
mass to enhance the growth of remaining trees (White 1980). Although
there is a strong economic incentive in forest management for wood
production (Puettman et al. 2015), there is an increasing understanding
how forest management can also be applied for supporting diversity of
forest structure (Bergeron et al. 2002, Kuuluvainen 2009), biodiversity
(Fedrowitz et al. 2014), resilience (Messier et al. 2013), and carbon
uptake (Hardiman et al. 2011). Additionally, there is an increasing
respect towards recreational opportunities and landscape amenities
(Butler \& Leatherberry 2004, Hugosson \& Ingemarson 2004, Urquhart \&
Cortney 2011). Therefore, structural diversity has been identified as a
silvicultural principle that could be addressed with non-conventional
forest management practices (Puettmann et al. 2015). Furthermore, forest
structural variety can be used as a measure for biodiversity and
structurally complex forests enhance carbon uptake (Gough et al. 2019),
which are important ecosystem services provided by forests. Thus, it can
be expected that structural diversity can also be beneficial for
functional and species diversity of forests.

\textbf{Conclusions}

Structural diversity at different scales can be linked to biodiversity
and carbon uptake as well as attractiveness of landscapes and
recreational activities. Possibilities for forest management in
considering these more varied objectives are acknowledged but objective
measures of structural diversity have been lacking as 3D information on
structure of forests and trees has practically been unavailable before
various laser scanning sensors. This study provides an example how
structural complexity of individual trees can be quantitatively assessed
and how it is affected by forest management. We demonstrated the use of
so-called box dimension in characterizing structural complexity of
individual Scots pine trees in managed plots. Thinning intensity
affected structural complexity and intensive thinning resulted in
increased structural complexity. Increasing crown size (i.e. width,
surface area and volume) and tree growth also increased structural
complexity but there was no relationship between structural complexity
and benefit-to-cost ratio (i.e. relationship between crown surface area
and stem volume) or light availability (i.e. competition). More research
is required to better understand the specific drivers behind structural
complexity of different tree species. For example, how regeneration and
growth of other than traditional attributes (e.g. DBH and height) affect
the structural complexity. It can be concluded that thinning had an
effect on structural complexity of individual Scots pine trees and stem,
crown, and growth attributes were identified as drivers for the
structural complexity in managed boreal forests.

\textbf{Acknowledgements}

The study was funded through the post-doctoral projects ``The effects of
stand dynamics on tree architecture of Scots pine trees'' (project
number 315079) and ``Capturing tree canopy water dynamics over time --
from single canopies to landscape and global dynamics'' (project number
330422) by the Academy of Finland. H.V. and K.C. were funded by BELSPO
(Belgian Science Policy Office) in the frame of the STEREO III programme
-- project 3D-FOREST (SR/02/355). K.C. was funded by the European
Union's Horizon 2020 research and innovation programme under the Marie
Sklodowska-Curie grant agreement No 835398.

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\textbf{Supplementary material}

Figure S1. Relationship between box dimension and diameter at breast
height (DBH) (A), tree height (B), and stem volume (C) grouped by
thinning treatment.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}

\end{center}
\end{figure}

Figure S2. Regression lines and scatterplots of box dimension against
surface-to-volume ratio (a proxy for a relationship between
photosynthetically active surface and building costs of a tree) of Scots
pine trees grouped by thinning treatment.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}

\end{center}
\end{figure}

Figure S3. Relationship between box dimension and growth of diameter at
breast height (DBH) (A), height (B) and stem volume (C) as well as
change in height-DBH ratio (\selectlanguage{greek}Δ\selectlanguage{english}H/DBH) (D) grouped by thinning treatment.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}

\end{center}
\end{figure}

Figure S4. Regression lines and scatterplots of Hegyi's competition
index against box dimension of Scots pine trees grouped by thinning
treatment.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}

\end{center}
\end{figure}

Figure S5. Relationship between box dimension and crown width (A), crown
projection area (B), and crown volume (C) grouped by thinning treatment.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image11/image11}

\end{center}
\end{figure}

\selectlanguage{english}
\FloatBarrier
\end{document}

