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Author(s): Vladimir Shanin, Hannu Hökkä and Pavel Grabarnik 
Title: Testing the Performance of Some Competition Indices against 
Experimental Data and Outputs of Spatially Explicit 
Simulation Models 
Year: 2021 
Version: Published version 
Copyright:   The Author(s) 2021 
Rights: CC BY 4.0 
Rights url: http://creativecommons.org/licenses/by/4.0/ 
 
Please cite the original version: 
Shanin, V.; Hökkä, H.; Grabarnik, P. Testing the Performance of Some Competition Indices against 
Experimental Data and Outputs of Spatially Explicit Simulation Models. Forests 2021, 12, 1415. 
https://doi.org/10.3390/f12101415 
Article
Testing the Performance of Some Competition Indices against
Experimental Data and Outputs of Spatially Explicit
Simulation Models
Vladimir Shanin 1,2,* , Hannu Hökkä 3 and Pavel Grabarnik 1






Citation: Shanin, V.; Hökkä, H.;
Grabarnik, P. Testing the Performance
of Some Competition Indices against
Experimental Data and Outputs of
Spatially Explicit Simulation Models.
Forests 2021, 12, 1415. https://
doi.org/10.3390/f12101415
Academic Editor: Rafał Podlaski
Received: 16 September 2021
Accepted: 12 October 2021
Published: 16 October 2021
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in
published maps and institutional affil-
iations.
Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
This article is an open access article
distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
4.0/).
1 Institute of Physicochemical and Biological Problems in Soil Science, Pushchino Scientific Center for Biological
Research of the Russian Academy of Sciences, 142290 Pushchino, Russia; pavel.grabarnik@gmail.com
2 Center for Forest Ecology and Productivity of the Russian Academy of Sciences, 117997 Moscow, Russia
3 Natural Resources Institute (LUKE), 00790 Helsinki, Finland; hannu.hokka@luke.fi
* Correspondence: shaninvn@gmail.com
Abstract: Three competition indices were tested against experimental data on the growth of in-
dividual trees in mapped forest stands and outputs of spatially explicit, process-based models of
competition. The comparison showed the fundamental importance of taking into account the spatial
structure of stands and, particularly, the relative spatial locations of individual trees (spatial asym-
metry) when calculating the competition between trees. Although none of the competition indices
are able to take into account the specific processes affecting the development of individual trees,
these indices can be used in forest dynamics modeling as a simplified representation of competition
between trees for resources.
Keywords: competition indices; spatial asymmetry; spatially explicit models; process-based models;
boreal forest stands
1. Introduction
The spatial structure of natural forest stands is usually complicated and defined by
the location of individual trees, their size, and species-specific peculiarities. The dynamics
of the spatial structure is driven by both external (various disturbances, including man-
agement) and internal (growth of trees, regeneration, competition-induced, and natural
mortality) factors. It is a common view that local interactions among individuals are a sub-
stantial factor for the formation of plant communities [1], and the quantitative estimation of
such interactions continues to be one of the most important research questions [2,3]. Impor-
tant features of competition that need to be accounted for in the forest ecosystem models
are species-specific strategies in the development of above- and belowground organs. Addi-
tionally, the plasticity of crowns and root systems in response to competitive pressure from
other individuals and heterogeneity of resources should be simulated with such models.
These features result in a complex pattern of interactions in plant communities, which
affect the sustainability and efficiency of the use of resources [4].
In order to estimate the between-tree competition, a large number of competition
models were developed, from very simplistic ones to spatially explicit multi-dimensional
mechanistic models [5–7]. Along with the mechanistic models of competition in plant
communities, the simple and common approach is the use of competition indices as
proxies for the intensity and direction of interactions between individual plants [8]. These
indices aim to quantify the resource availability for individual trees taking into account the
interactions between trees. The importance of assessment of competition tension is related
to the necessity of correct estimation of the mortality rate of trees due to self-thinning. The
competition indices are built on the basis of morphometric features of trees and are usually
functions of distance to the closest neighbor trees and their characteristics. Later, the spatial
structure of stands was also taken into account using the angular approach to assess the
Forests 2021, 12, 1415. https://doi.org/10.3390/f12101415 https://www.mdpi.com/journal/forests
Forests 2021, 12, 1415 2 of 11
influence of neighbor trees [9]. Alternatively, an approach based on the estimation of
overlap of influence zones has been proposed [10]. The competition indices are widely
used due to their simplicity, in comparison to more detailed spatially explicit mechanistic
models of competition based on the description of processes underlying the development
of tree crowns and root systems and resource acquisition, which are thus capable to take
into account the effect of spatial locations of trees on the strength of interactions between
them [11,12]. Such models usually have a complex structure and a higher number of
parameters, and therefore demand a considerable computational effort. Therefore, finding
the competition index that provides comparable accuracy of estimation of the number
of resources allocated among trees in complex forest stands will allow developing more
efficient ecological models, especially for planning in uneven-aged forestry, where the
competition among trees and possible changes due to cuttings play the most crucial role in
growth and survival of trees.
The purpose of the study was to analyze how the capability of competition indices to
take into account the relative spatial locations of trees affects their performance in repro-
ducing the interactions between trees. We compared the performance of three competition
indices with outputs of spatially explicit mechanistic models of above- and belowground
competition, as well as with the empirical data. To apply the indices to multispecies forest
stands, the species-specific parameters were estimated based on the independent dataset
for Scots pine (Pinus sylvestris L.), Norway spruce (Picea abies (L.) H. Karst.), birch (the
same parameters were set for both silver birch Betula pendula Roth and downy birch Betula
pubescens Ehrh.), and European aspen (Populus tremula L.), hereafter referred to as pine,
spruce, birch, and aspen, respectively.
2. Materials and Methods
2.1. Description of Competition Indices
We took for the comparison the indices that utilize standard data of forest surveys (e.g.,
stem height and/or diameter at breast height (DBH)) and have species-specific coefficients.
The Hegyi index [13] CIHi estimates competition on the basis of the size of focal tree i,
that of neighbor tree j, and also the distance between them as follows:
CIHi =
n
∑
j 6=i
DBHj
DBHi · rij , (1)
where DBHi is the stem diameter at breast height of focal tree i (cm), DBHj is the stem
diameter at breast height of neighbor tree j (cm), rij is the distance between focal tree i and
neighbor tree j (m), and n is the number of neighbors of the focal tree i.
The selection of neighbor trees for the calculation of the index is based on the calcula-
tion of angles γi and γj. Angle γi is calculated as follows:
γi = 2 · arctan
(
DBHi
A
)
, (2)
where A is the species-specific parameter defining the search radius (m) (Table 1), and
DBHi is the stem diameter (cm) at breast height for focal tree i.
Table 1. Species-specific coefficients used for the calculation of competition indices.
Species A γl x y z W α β
pine 4.5 5.0 5 4 3 18 3.64 0.95
spruce 3.0 16.0 4 1 17 15 4.07 0.29
birch 5.5 11.5 3 5 7 19 3.82 1.22
aspen 5.0 10.9 3 5 5 21 3.78 1.25
Forests 2021, 12, 1415 3 of 11
Angle γi is to be compared with angle γj, which is calculated as follows:
γj = 2 · arctan
(
DBHj
rij
)
, (3)
where DBHj is stem diameter (cm) at breast height for neighbor tree j, and rij is the distance
between focal tree i and certain neighbor tree j (m). All trees meeting the conditions that
γi ≤ γj and rij ≤ A are considered as neighbor trees for the given focal tree i.
Here, we reintroduce the angular competition index CIAi, which was initially suggested
in [14]. The specific feature of this index is based on the angle sectors formed by tangent
lines from the location of the focal tree to the edges of the influence zones of its neighbor
trees. The neighbor selection criterion is similar to those for the Hegyi index (Equation (2)),
but the angle γi is compared with the certain species-specific threshold value of γl (Table 1).
After the selection of competing trees for each neighbor tree j, the radius Rj of its
influence zone on the focal tree i is calculated.
Rj =
(
Ari
c
) 1
x ·
(DBHj
DBHi
)y
, (4)
where x, y, c are scaling factors, DBHi and DBHj are the stem diameters at breast height of
the focal and neighbor trees, respectively, and the index of crown projections overlap Ari is
calculated as follows [15]:
Ari =
(
∑nj=1 aij −∑k−1f=1 ai f +∑mg=1 aig
)
+ CAi
CAi
, (5)
where n is the number of trees that have overlapping crown projections (represented as
circles) with the crown projection of the focal tree i, k is the number of trees for which the
crown projections completely overlap with the crown projection of the focal tree i, m is the
number of trees for which the crown projections are inside the crown projection of the focal
tree i, and CAi is the theoretical crown projection area of tree i for the case of growth in
open space, which is calculated as follows:
CAi = pi ·
(
W · DBHi
2
)2
, (6)
where W is the species-specific ratio of crown projection diameter to the stem diameter at
breast height of the tree, and DBHi is the stem diameter at breast height of the focal tree.
The crown projection radius CRi is calculated in a similar way.
The area of overlapping crowns projections of trees i and j, aij, is calculated as follows:
aij =
CR2i · (Fi − sin(Fi))
2
+
CR2j ·
(
Fj − sin
(
Fj
))
2
, (7)
where
Fi = 2 · arccos
(
CR2i − CR2j + r2ij
2 ·CRi · rij
)
, (8)
Fj = 2 · arccos
(
CR2j − CR2i + r2ij
2 ·CRj · rij
)
, (9)
where CRi, CRj are the radii of crown projections of the focal and neighbor trees, respectively,
and rij is a distance between the trees.
Further, the sectors were determined by tangent lines from the focal tree location to
the edges of the influence zones of each neighbor tree Rj. The resulting competition index
is the arithmetic mean of two components, the first of which represents the belowground
Forests 2021, 12, 1415 4 of 11
competition, and the second one represents the aboveground competition (assuming the
equal influence of both kinds of competition on the growth of trees). The first component
is calculated as the sum of angular sizes of sectors, while for the calculation of the second
component the overlapping sectors are excluded (i.e., overlapping areas of sectors are
accounted for only once). To account for the heterogeneity of the incoming photosyn-
thetically active radiation (PAR) from different directions, an additional weighting factor
was introduced. For azimuth 0◦ (geographic north) this factor was set at 0.5, for azimuth
180◦ (south), it was set at 1.5, for azimuths 90◦ and 270◦ (east and west, respectively), it
was set at 1.0, and for intermediate directions, the value of this coefficient is interpolated
between the closest base values with 0.1◦ discretization. The values of the scaling factor
were estimated from the parameterization of the spatially explicit model of aboveground
competition [12]. The schematic representation of the calculation of the angular index is
provided in Figure 1.
Figure 1. The schematic representation of the calculation of the angular competition index. Influence
zones of neighbor trees N1–N6 generate sectors of the influence zone of the focal tree F. Combining
these sectors with and without taking into account the overlapped areas produces the components of
the competition index related to the root and crown competition, respectively. The intensity of gray
filling denotes the competition strength.
By construction, the angular completion index takes into account the relative locations
of neighbor trees. This property, which can be attributed to spatial asymmetry, is seldom
taken into account in many of the known competition indices.
The third competition measure in the comparison study is based on the competition
kernels CIKi which determine the competitive load not at locations of trees specifically
but rather at each point of a plot area, and thus, they are the alternative for competition
indices [16]. Generally, they can be represented as function pi(z), which describes the local
influence of a certain tree i on its environment at a point z [17].
pi(z) =
DBHαi
1 +
(
ri(z)
β
)2 , (10)
where DBHi is stem diameter at breast height, ri(z) is the distance between the focal tree
and given point, α and β are species-specific coefficients (Table 1).
To calculate the competitive pressure on each tree in a forest stand, the influences of
all neighbor trees in the point corresponding to the location of the focal tree were summed
as follows:
CIKi = ∑
j 6=i
pj(i), (11)
Forests 2021, 12, 1415 5 of 11
where pj(i) denotes the influence of the neighbor tree j on the focal tree i.
In contrast to competition indices, which describe the influence of neighbor trees only
in the point corresponding to the location of the focal tree [18], the competition kernels form
a competition field. This, in theory, could be useful for the assessment of the influence of
trees on the undergrowth and ground-layer vegetation.
Since the competition effect is also dependent on the size of the focal tree (expressed
through its stem diameter at breast height, DBHi), the competition index was transformed
as follows:
CIKi =
CIKi
DBHi + CIKi
. (12)
According to [18], such transformation increases the biological plausibility and im-
proves the model behavior due to the nonlinear relationship between the size of the focal
tree and the weakening of the competitive pressure.
The calculation of all competition indices was carried out in statistical programming
environment R v. 3.1.1 [19]. The values of coefficient W (Table 1) were estimated from
the experimental data on the crown size on the sampling plots [20,21]. The values of
other coefficients were estimated with the least-squares method using the data from the
yield tables [22,23]—namely, the parameters of the equations for competition indices were
estimated by fitting to the normalized values of annual volume increment of pure stands
reported in these tables.
2.2. Brief Description of Spatially Explicit Mechanistic Models of Competition
Two simulation models applied in the current study are mechanistic, individual based,
and spatially explicit. Each tree has an explicitly defined location on the simulation plot,
which is divided into square cells. The multi-layered model of belowground competition for
available nitrogen in soil [11] describes the spatial distribution of belowground biomass of
trees. The area of the nutrition zone is calculated based on the mean distance of horizontal
spreading of roots, which, in turn, depends on DBH and species-specific coefficients. The
preferability of occupation of a certain cell is dependent on the amount of available nitrogen
in this cell, the distance between cell and stem base, and the biomass of fine roots of other
competing trees in the given cell. The vertical distribution of biomass of roots is described
by function with species-specific parameters. The intensity of nitrogen uptake is dependent
on species-specific features, the age of a tree, and the biomass of its fine roots in a given
cell. The main output from the model is the amount of nitrogen consumed by each tree.
The model of aboveground competition allows calculating the amount of PAR captured
by each tree. An advanced property of the model is its capability to simulate the crown
asymmetry arising as an adaptive response of a tree on the competitive pressure from
neighbor trees [12]. The crown shape is species specific, while crown dimensions are
dependent on the height and DBH of the tree, and the local stand density. The crown of a
tree is approximated by 3-dimensional cells, and crowns of several trees can be presented in
the same cell. To take into account the influence of neighbor trees, the Voronoi partitioning
is applied to the simulation plot. Depending on the available space, determined by Voronoi
polygons, the crown width in different directions is modified, thus allowing the simulation
of asymmetry resulted from the inter-tree competition for light. The model also simulates
the distribution of biomass inside the crown using species-specific coefficients. At the
final stage, the model calculates the amount of PAR intercepted by the given tree with a
simplified ray-tracing technique [12]. The amount of intercepted PAR is affected by leaf
area index (LAI) in a given cell, while the transmitted part of PAR is passed to lower canopy
layers [24]. The main output from the model is the amount of PAR absorbed by each tree.
These two models have species-specific parameters describing the distribution of
roots in soil and foliage inside the crown, and also peculiarities of nitrogen uptake and
light interception. These features allow simulation of intra- and inter-specific competition
and represent the plasticity of root systems and crowns as an adaptation mechanism to
the competition for resources. Thus, the above-described models represent life strategies
Forests 2021, 12, 1415 6 of 11
of different tree species and take into account the spatial heterogeneity of the environ-
ment, which allows them to simulate more accurately the competition and tree growth
in heterogeneous forest stands. The models were implemented using the Object Pascal
programming language with Lazarus v. 1.2.4 Integrated Development Environment (Free
Pascal Compiler v. 2.6.4) under Debian GNU/Linux 8.
2.3. Design of the Simulation Experiment
We used a set of simulation scenarios to compare the competition indices by outputs
of spatially explicit models. All simulations were carried out on a simulation plot of 50 ×
50 m divided into square cells of 0.5 × 0.5 m. The plot was “wrapped around the torus” to
eliminate the edge effect [25].
The set of scenarios included a simulation of stands of different development stages
and spatial structures (Table 2). Simulated stands consisted of spruce, pine, birch, and
aspen in different combinations. Three types of spatial structure were simulated: regular
(location of trees on a regular grid), random, and location of trees in several dense clusters
according to Neyman–Scott algorithm [26]).
Table 2. Initial parameters (±standard deviation) of stands used for simulations.
Stand Type Age, Years Density, ha−1 Height, m DBH, cm
Young 5 10,000 1.8 ± 0.45 1 ± 0.25
Pole 30 1600 5 ± 1.25 5 ± 1.25
Mature 70 448 22 ± 3.3 25 ± 3.75
2-layered Canopy 70 224 22 ± 3.3 25 ± 3.75
Undergrowth 30 800 5 ± 1.25 5 ± 1.25
Uneven aged 20–230 768 1.9–19.2 4.1–38.9
To standardize calculated values of all three competition indices (CIH, CIA, and CIK),
they were transformed as follows:
CItransi = 1−
CIi
max(CI)
, (13)
where CIi denotes any type of competition index for the i-th tree.
By this transformation, we obtain a new set of values that fits the range [0, 1] where
the least value corresponds now to the tree received the minimal amount of resource (i.e.,
the most suppressed tree in a stand), and the maximal value corresponds to the least
suppressed tree. To estimate the sensitivity and make the conclusions justified, the Monte
Carlo runs with 1000 repetitions and 10% variation of parameters were used.
The amount of available nitrogen and absorbed PAR by each tree were calculated with
spatially explicit mechanistic models of below- [11] and aboveground competition [12],
respectively. The obtained values were similarly normalized to fit the range [0, 1].
REStransi =
RESi
max(RES)
, (14)
where RESi is the number of resources obtained by a certain tree. For each tree, the
arithmetic mean between normalized values of resources obtained from above- and below-
ground competition was calculated.
To estimate the agreement between outputs from the detailed mechanistic models of
competition and the simplified competition indices, we used the Pearson linear correla-
tion coefficient r. Calculations were carried out in statistical programming environment
R v. 3.1.1 [19].
Forests 2021, 12, 1415 7 of 11
2.4. Description of Experimental Data Used for Comparison of the Competition Indices
The comparison between competition indices was carried out also by the experimental
data on the growth of individual trees in mapped stands. The experimental dataset on peat-
land forests drained in 1963 was obtained from the Heinävesi experimental site (Heinävesi
municipality) in eastern Finland (62.45◦ N, 29.10◦ E, and 114 m a.s.l). The long-term
(1981–2010) mean annual precipitation is 575 mm, and the temperature sum of the growing
season is 1248 dd. (threshold value + 5 ◦C). The area represents predominately Norway
spruce forest with an admixture of Scots pine and downy birch classified as Vaccinium
myrtillus type drained peatland [27]. The site was divided into 12 subplots representing
different thinning treatments conducted in 1994, with a total area of approximately 2.6 ha.
Consequent inventories were carried out in 1994, 1998, 2003, and 2008, measuring the DBH
of each tree. Excluding trees that did not survive until the last inventory, the total number of
measured trees comprised 920 individual trees. The DBH ranged between 4.6 and 46.9 cm,
with mean value increased from 17.0 cm in 1994 to 21.5 cm in 2008. The distribution was
unimodal and almost symmetric (slightly left skewed). Due to the nonsquare shape of plots,
traditional methods of elimination of the edge effect were inapplicable. Instead, a buffer
area of 6 m on edge of each plot was chosen, and trees in this zone were excluded from the
analysis. Therefore, the total number of trees accounted for in the analysis comprised 277.
To estimate the performance of different competition indices and estimate the relation-
ship between 5-year diameter increment and values of different competition indices, we
applied the Spearman rank correlation coefficient, as suggested in [28]. The 5-year diameter
increment for the first period (1994–1998) was calculated by multiplying the original value
by (5/4).
3. Results and Discussion
The correlation analysis between outputs from detailed mechanistic models of competi-
tion and competition indices (Figure 2) showed that the angular competition index represents
the distribution of common resources among individual trees with the best similarity to
models of competition, among three considered indices. The mean value of correlation
coefficient (r) was 0.728, and for 70% of stands, the value of r for the angular competition
index was the highest among all three considered indices. For this index, the cases with
r < 0.5 were absent. The Hegyi index also showed satisfactory results: the mean value of r
was 0.612, and for 20% of cases r was less than 0.5. For 30% of cases, the value of r for this
index was the highest among all three considered indices. The competition kernels showed
a lower correlation, with the mean value of r at 0.556, and r < 0.5 in 28% of cases. Better
performance of angular-based indices was also shown previously [29].
The competition indices more efficiently represented the competition in young and
pole, rather than in mature stands. The considered indices showed the best agreement with
detailed models of competition in the case of random and regular placement of trees. Most
likely, such results can be explained by an underestimation of the adaptation of trees to
competitive pressure from neighbors through asymmetric development of crowns and root
systems [4,20,30–32].
The correlation analysis between different indices showed that the best similarity was
observed between the Hegyi index and angular index, where the value of r ranged between
0.692 and 0.939 (the mean value comprised 0.845). The range of r between the Hegyi index
and competition kernels across all sample stands ranged between 0.481 and 0.928, with a
mean value at 0.768. The comparison between competition kernels and angular index showed
lower values of r (minimum at 0.196, mean at 0.590, and maximum at 0.777). In general,
the correlation between values of different indices decreased with the age of the stand.
Forests 2021, 12, 1415 8 of 11
Figure 2. Pearson linear correlation (r) between outputs from three competition indices (CIi) and
outputs from mechanistic spatially explicit models (RESi). Dashed horizontal line denotes r = 0.5.
Stand type codes are the following: uppercase letters denote species (P—pine, S—spruce, B—birch,
A—aspen), first lowercase letter denotes stand age (y—young, p—pole, m—mature, t—two layered,
u—uneven aged), and second lowercase letter denotes spatial structure of stand (r—random location
of trees, c—clustered location, g—location on a regular grid). The standard deviation obtained with
the Monte Carlo runs is also shown.
Figure 3 represents the relationship between different competition indices for some sample
stands with more details. The general trend is the increase of difference between indices with
the stand age, thus reflecting the increased role of spatial heterogeneity. The relationship
between the Hegyi index and angular index is linear, while it is nonlinear for other cases.
The correlation between 5-year DBH increment and values of competition indices
(Table 3) generally showed better performance of the angular index and the similar perfor-
mance of Hegyi index and competition kernels. Lower correspondence between competition
indices and measured diameter increment, in comparison to the correspondence between
competition indices and outputs of competition models, can also originate from our as-
sumption on the equal contribution of above- and belowground competition, while in real
stands, these types of competition may differ in their importance for the performance of
individual trees.
Table 3. The values of Spearman correlation coefficients between the 5-year tree diameter increment
and three different competition indices for 12 sample subplots of the Heinävesi experimental site.
The values > 0.5 in absolute magnitude are shown in bold.
Subplot Hegyi Index Angular Index Competition Kernel
1 0.544 0.657 0.493
2 0.590 0.662 0.515
3 0.567 0.669 0.509
4 0.556 0.680 0.499
5 0.545 0.677 0.516
6 0.567 0.671 0.520
7 0.571 0.663 0.516
8 0.559 0.672 0.515
9 0.584 0.665 0.507
10 0.539 0.652 0.489
11 0.590 0.679 0.520
12 0.548 0.674 0.509
Forests 2021, 12, 1415 9 of 11
Figure 3. The normalized values of different competition indices for some sample stands, plotted
against each other. The upper row represents the young pine stand with random placement of trees,
the middle row represents the pole stand with equal portions of birch and aspen with a clustered
placement, and the bottom row represents the mature pine stand with spruce undergrowth under
the canopy.
4. Conclusions
The comparison with outputs from spatially explicit mechanistic models of above-
and belowground competition showed the fundamental importance of taking into account
the spatial structure of stands and, particularly, the relative positions of individual trees
when calculating the competition between them. Such importance is supported by the best
results shown by the angular index.
The angular competition index is relatively simple and can be calculated from the basic
and easily measured characteristics of trees, such as stem diameter or height, and in the case
if their locations are known. Moreover, it allows for estimating the competition based on the
differences between above- and belowground competition, while most of the competition
indices were designed and parameterized based on aboveground competition only.
However, comparison with measured data on the growth of individual trees showed
that any competition index could be considered only as a simplified representation of
competition between trees for resources because it is unable to take into account all possible
causes that might affect the development of each tree. Nevertheless, due to the simplicity
of competition indices, they can be useful in cases where the detailed mechanistic models
are inapplicable. For example, the models based on competition indices can be applied for
individual-based simulation of stand development at large scales with low computational
demands. The competition indices, due to their simplicity, can be also used for the analysis
Forests 2021, 12, 1415 10 of 11
of how the parameters of trees and the spatial structure of stand affect the interaction
between trees.
The tree species that were chosen for the analysis are typical for the European boreal
forests. The open question is if the study outputs are applicable for other forest zones, e.g.,
for temperate or tropical forests, where the limiting factors are different. However, we
would expect that the general findings in spite of the different representation of interactions
between trees of various life strategies will be valid irrespective of forest type and tree
species and will be determined by the spatial structure of the stand. Another possible
limitation is that our approach was not tested in relation to the changes in forest ecosystems
that resulted from either natural development or disturbances, including climate change
or forest management activity. Since the indices are calculated on the basis of the relative
positions and sizes of individual trees, any changes related to the stand dynamics will be
represented with the values of indices. However, in the case of environmental changes
affecting ecosystem parameters not accounted for in the indices, the approach would
require additional testing and additional data.
Author Contributions: Conceptualization, V.S. and P.G.; methodology, V.S. and P.G.; software, V.S.;
validation, V.S.; formal analysis, V.S.; investigation, V.S. and P.G.; resources, H.H.; data curation, H.H.;
writing—original draft preparation, V.S.; writing—review and editing, P.G. and H.H.; visualization,
V.S.; supervision, P.G.; project administration, P.G.; funding acquisition, P.G. and H.H. All authors
have read and agreed to the published version of the manuscript.
Funding: The study was supported by the Russian Science Foundation, project #18-14-00362
(Vladimir Shanin, Pavel Grabarnik–constructing and testing the indices, APC payment), and by
the Academy of Finland, projects #322972, #312912, and #336570 (Hannu Hökkä–collecting and
processing the experimental data for validation).
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data are contained within the article.
Acknowledgments: Not applicable.
Conflicts of Interest: The authors declare no conflict of interest.
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