sustainability
Article
Height Allometry of Pinus nigra Arn. in Troodos National
Forest Park, Cyprus
Dimitrios I. Raptis 1, *, Vassiliki Kazana 1 , Nikolaos Onisiforou 1 , Christos Stamatiou 1
1
2
*
Citation: Raptis, D.I.; Kazana, V.;
Onisiforou, N.; Stamatiou, C.;
and Angelos Kazaklis 2
Department of Forest and Natural Environment Sciences, International Hellenic University,
66100 Drama, Greece; vkazana@for.ihu.gr (V.K.); nikosonisiforou12@hotmail.com (N.O.);
stamatioy@gmail.com (C.S.)
OLYMPOS-Centre for Integrated Environmental Management, 55132 Thessaloniki, Greece; akaz98@otenet.gr
Correspondence: d_rapt@for.ihu.gr; Tel.: +30-25210-60460
Abstract: Total height is one of the basic morphometric tree variables included in all forest management inventories, because it is connected with several forest processes and functions related to the
estimation of the woody tree volume and sustainable forest yield. The current research, based on a total sample of 1059 measured Black pine (Pinus nigra Arn.) trees from Cyprus, is an attempt to explore
the biological processes related to the tree height allometry of this species and develop a generalized
mixed-effects model for tree height prediction. The proposed model, with three additional basic
covariates and two random parameters, explained almost 96% of the height variance. The model
results showed that although competition and site-connected variables affected the total height, it
was the crown base height that explained, to a large degree, the height expression. Through the
mixed-effects modeling approach it was possible to explore the complex biological processes related
to the tree allometry of Black pine and depict those within a mathematical formulation. The proposed
generalized model decreased the error significantly, and therefore it can be used for operational forest
management purposes. However, for marginal predictions, use of only the fixed part of the basic
model could suffice, since this also provided unbiased parameter estimates.
Kazaklis, A. Height Allometry of
Pinus nigra Arn. in Troodos National
Keywords: forest ecosystems; Näslund function; pine ecology; tree structure
Forest Park, Cyprus. Sustainability
2021, 13, 5998. https://doi.org/
10.3390/su13115998
Academic Editor: Ivo Machar
Received: 8 April 2021
Accepted: 18 May 2021
Published: 26 May 2021
Publisher’s Note: MDPI stays neutral
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Copyright: © 2021 by the authors.
Licensee MDPI, Basel, Switzerland.
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distributed under the terms and
conditions of the Creative Commons
Attribution (CC BY) license (https://
creativecommons.org/licenses/by/
1. Introduction
Total height is one of the basic morphometric tree variables. It directly affects
the woody tree volume and sustainable forest yield, and it is currently included in all
forest management inventories and plans. It has been connected to the site index [1],
light capture [2], growth dynamics and succession [3], carbon sequestration [4] and tree
stability [5]. From an ecological perspective, total height is a fundamental component of
processes and functions at a tree or at an ecosystem level [6]. It reflects the outcome of the
combined effect of several endogenous and exogenous variables that may be expressed
through complex models for height prediction.
The diameter at breast height is a key element in forest inventories [7] and the main
covariate in linear or nonlinear basic models for height prediction. However, the height–
diameter relationship deviates between different forest stands [8]. In order to solve this
problem, additional covariates were introduced in the proposed models, thus leading to
the development of models suitable for regional use [9]. Temesgen and von Gadow [10]
introduced the term “generalized” to describe the height–diameter curve with additional
variables at the stand level. The generalized models have improved the height prediction accuracy for large forests, and they have allowed the description of the ecological
connections between the variables included in the models. Through this, the researchers
managed to explore the ecological processes behind the height expression at various levels,
providing ecological explanations for the largest part of height variations [11].
4.0/).
Sustainability 2021, 13, 5998. https://doi.org/10.3390/su13115998
https://www.mdpi.com/journal/sustainability
Sustainability 2021, 13, 5998
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The development of generalized models is currently based on two approaches [12].
According to the first approach, the parameters of the basic models are expressed using
mathematical forms of the stand variables [13]. Usually, a preliminary comparison between
the height and the independent variable(s) is performed prior to the model fitting. The
second approach recommends the direct input of the additive covariates into the model [12].
However, regardless of the approach followed, the implementation of the generalized
models is not fully adequate due to the fact that similar stand characteristics assume similar
height–diameter curves [14]. Moreover, the inclusion of a large number of independent
variables may lead to over-parameterization problems.
The mixed-effects approach provided new perspectives on height modeling
procedures [15]. These types of models are recommended when analysis should be carried
out on nested data from tree height observations obtained from installed sample plots,
which usually present a lack of independence between them [16]. Mixed-effects models
are mainly composed of two parts, the fixed-effects part that refers to the properties of
the entire population and the random-effects part that is associated with the individual
properties of separate sample plots [17]. Their random part may also capture the effect of
independent variables that are not included in the fitted model [18]. Thus, the mixed-effects
modeling technique is a powerful tool for the development of generalized models and the
ecological exploration of processes and functions behind the tree height allometry.
The basic aims of the current study were as follows: (i) the development of a generalized mixed-effects model for the tree height prediction of Black pine (Pinus nigra Arn.)
and (ii) the exploration of the ecological process of each covariate included in the proposed
model of Black pine height allometry. For the model development and evaluation in terms
of fitting ability, field data derived from the natural even-aged Black pine forests of Troodos
National Forest Park in Cyprus were used. In this region, Black pine forms a separate
population and is characterized by geographical isolation. Despite the fact that it has been
studied both in Turkey [19] and in Greece [20], the Cyprus populations have never been
studied before and they constitute the focus of the current study.
2. Materials and Methods
Black pine, a native pine species of the Troodos Mountains in Cyprus, is often characterized as a mountain pine in the wider region. At lower altitudes it is intermingled with
Calabrian pine (Pinus brutia Ten.) and dominates above 1400 m of elevation [21], forming
pure and naturally regenerated even-aged stands. These types of forests cover a total area
of 9147 hectares at the center of the south and south-west parts of the island (Figure 1).
In higher locations, the Black pine is intermingled with Juniper (Juniperus foetidissima
Willd.) as a secondary species. The soil is shallow, residual and skeletal from the underlying
diabase and gabbro, becoming stiffer in mild slopes [22]. The meteorological data from
the nearest station in Prodromos area indicate that the mean annual precipitation is about
790 mm, and the mean annual temperature is 13 ◦ C.
2.1. Sampling
The spatial information on the Black pine distribution in the Troodos Mountains
was provided by the Department of Forests of the Cyprus Ministry of Agriculture, Rural
Development and Environment in a GIS file format (shapefile). Using the “create random
points” module in a GIS environment, 30 potential locations for field measurements were
identified. However, only 28 sample plots were installed in the field, since access in
two of the locations was not feasible due to physical obstacles. The sample units (plots)
were circular, with a radius of 12.62 m in a horizontal projection, covering a total area of
500 m2 each. Within each plot, the diameter at breast height (1.30) (dbh) was measured
with a caliper.
The total height (h) and the crown base height (cbh) were measured using a LaserAce
hypsometer (Trimble, Sunnyvale, CA, USA). The crown width (cw) was measured using
the “canopy spread” module of the same instrument in two vertical directions. The total
sample included 1059 Black pine trees and the descriptive statistics of the main measured
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variables are presented in Table 1. Trees with damaged or broken tops were excluded from
the sample.
Based on the measurements at tree level, several stand-based indexes were estimated
covering basic aspects of the soil productivity and competition regimes within the forests
stands. The dominant height (hdom ) and the dominant diameter (ddom ) were calculated
from the mean values of the 100 thickest trees per hectare or equally proportional to the
plot size [23]. The estimated density indexes included the basal area per hectare (m2 ·ha−1 ),
the number of trees (stems) per hectare (N·ha−1 ), the quadratic mean diameter (qmd),
the stand density index (sdi) [24], the relative spacing index (rsi) [25] and the spacing
coefficient (sc) [26,27]. Their mathematical expressions are as follows:
qmd
sdi = n
25.4
rsi = √
1.605
100
n hdom
100
sc = √
n cw
(1)
(2)
(3)
where n is the number of the estimated stems per hectare at plot level. The crown competition factor (ccf) [28] was calculated using a quantile regression for a nonlinear mixedeffects model of the maximum crown width from the 95th percentile points, based on the
qrNLMM package [29] in the R (cran) environment. The fitted model was of the simple
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power form y = axb according to the findings of Raptis et al. [18] for Black pine in Greece,
where y is the maximum crown width, x is the diameter and a, b are the fixed parts of the
mixed-effects model.
Figure
Figure1.1.The
TheTroodos
TroodosMountains
Mountainsstudy
studyarea
areaand
andthe
thespatial
spatialdistribution
distributionofofthe
thesample
sampleplots.
plots.
2.1. Sampling
The spatial information on the Black pine distribution in the Troodos Mountains was
provided by the Department of Forests of the Cyprus Ministry of Agriculture, Rural Development and Environment in a GIS file format (shapefile). Using the “create random
points” module in a GIS environment, 30 potential locations for field measurements were
identified. However, only 28 sample plots were installed in the field, since access in two
of the locations was not feasible due to physical obstacles. The sample units (plots) were
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Table 1. Descriptive statistics of the data used for model development.
Troodos National Park (n = 1059)
diameter (cm)
height (m)
crown width (m)
crown base height (m)
stems per hectare (n)
dominant height (m)
dominant diameter (cm)
basal area (m2 ·ha−1 )
quadratic diameter (cm)
stand density index
relative spacing index
spacing coefficient
crown competition factor
Mean
Min
Max
St. Dev.
29.31
14.81
3.77
10.16
756
20.66
49.49
31.35
23.30
637
0.18
0.97
184
3.30
1.30
0.60
0.70
1360
14.02
37.00
14.66
17.16
322
0.14
0.78
96
74.60
29.80
9.60
22.20
460
25.88
64.20
56.87
34.18
999
0.26
1.09
273
14.04
5.93
1.68
4.74
193
2.77
7.13
8.41
4.42
137
0.02
0.08
37
2.2. Statistical Analysis
2.2.1. Basic Model Selection
For the basic model selection, 5 fundamental nonlinear models for height–diameter
allometry were selected for evaluation.
The selected models (Table 2) included only 2-parameter nonlinear forms due to
convergence problems experienced during model fitting. Taking into consideration the
hierarchical nested structure of the available data, an OLS fitting would lead to underestimates of the covariance matrix of the parameter estimates and the residual variance of
the regression equation [16]. Due to this limitation, instead of an OLS fitting procedure, it
was decided mixed-effects modeling at two different levels would be used, for the simple
model selection and for the generalized expanded form. A detailed description of the
nonlinear mixed-effects modeling technique is provided by Calama and Montero [30]. All
the model parameters were assumed to contain a random part and their variances were
reported. On condition of almost zero variance, the parameter was regarded as fixed. In
case of convergence problems, the mixed parameters were reduced gradually [17].
Table 2. 2-Parameter models and mathematical forms of mixed-effects.
Model Number
Basic model
Form
h = 1.3 +
Reference
2
dbh
(β1 dbh+β0 )2
[31]
2
M1
dbh
hi = 1.3 + [(β +b )dbh
+(β0 +b0i )]2
1i
1
Basic model
h = 1.3 + (ββ0+dbh
1 dbh)
M2
hi = 1.3 + (β 0+b 0i+dbh)
1i
1
Basic model
h = 1.3 + β0 [1 − exp(−β1 dbh)]
M3
hi = 1.3 + (β0 + b0i )[1 − exp(−(β1 + b1i )dbh)]
Basic model
h = 1.3 + 10β0 dbhβ1
M4
hi = 1.3 + 10(β0 +boi ) dbh(β1 +b1i )
Basic model
h = 1.3 + β0 dbhβ1
M5
hi = 1.3 + (β0 + b0i )dbh(β1 +b1i )
[32]
(β +b )dbh
[33]
[34]
[35]
Where β0,1 is the fixed part of the model, b0,1 is the random part in i plot, h is the total height and dbh is the
diameter at breast height.
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The selection of the basic model was based on the following five criteria of good fit:
(i) the Root Mean Square Error (RMSE); (ii) the Fitting Index (FI) that was based on the R
squared mathematical expression [36]; (iii) the Akaike information criterion [37], which
penalizes the number of the model’s parameters; (iv) the Bayesian information criterion,
which also penalizes the total number of the model’s parameters [38] and (v) the mean
prediction bias. Their mathematical expressions are as follows:
RMSE =
v
u
u ∑n h − ĥ 2
j
t j=1 j
n−p
(4)
2
∑nj=1 hj− ĥj
FI = 1 −
2
∑nj=1 hj − hj
(5)
AIC = −2 log(L) + 2p
(6)
BIC = −2 log(L) + plog(n)
∑nj=1 hj − ĥj
Bias =
n
(7)
(8)
where hj , ĥj and hj represent the measured, estimated and mean values of the total height
variable of the jth observation, respectively, n represents the number of observations
forming the total sample, p represents the number of estimated model parameters and L
represents the likelihood of the fitted model.
In case of an unequal error variance, a function to regulate it was applied in a power,
exponential or constant/power mathematical form by linking the variance with a predictor.
Assuming that the i indicator represents the different plots and the j indicator each tree
within the plot, the variance functions that were tested were the following:
Var εij = σ2 x2δ
(9)
ij (power function)
Var εij = σ2 exp 2δxij (exponential function)
2
Var εij = σ2 δ1 + xδij 2 (constant-power function)
(10)
(11)
where δ, δ1 and δ2 are the parameters to be estimated and x is the independent predictor to
stabilize the variance of the within-plot heteroscedasticity. Since dbh has been successfully
used as an independent predictor to regulate the variance in relevant research works for
Black pine [18,39], it was decided that the same method would be followed. The most
suitable function along with the most appropriate predictor was determined by visual
inspection of the residuals and the results of the likelihood ratio test (LTR), [40] the AIC
and the BIC criteria, as these have been incorporated in the anova function for nested
models [17]. According to the same authors [17], the anova function, when two or more
arguments of fitted models are used, produces likelihood ratio tests, which compare the
models and provide the p value from the χ2k2 −k distribution, for k2 −k1 degrees of freedom,
1
with ki representing the number of parameters to be estimated in model i. In order to test
the normality assumption of the residuals, the graphical Q–Q plot method was used.
2.2.2. Generalized Model Expansion
The selected model was the basis for developing a reliable generalized model and
explaining height allometry on an ecological basis in the second phase. The development
of the generalized model was based on the correlation between the plot-specific parameters
of the simple model and the variables at stand level [23]. The estimation of the plot-specific
parameters was based on the nlsList function of the nlme library. It is an alternative method
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for model expansion, closely linked to mixed-effects modeling. Through this procedure,
specific covariates at tree and stand level were incorporated in the expanded generalized
model in order to reduce the random variance of the simple model parameters.
At the final stage, the generalized form was expanded to a generalized mixed-effects
model in order to explain the largest possible portion of height variance. Again, all the
parameters were assumed to contain a random part [41] as long as convergence was
attainable. Otherwise, the number of the mixed parameters was reduced gradually until
convergence was achieved. In a case of convergence, the mixed parameters were further
reduced, and the nested models were compared using the anova function. In a case of no
significant difference between the nested models (likelihood ratio statistic p-value > 0.05),
the simplest form was selected [17]. Finally, the graphical solution was preferred to evaluate
the fitting performance of the selected model against the total dataset of height–diameter
observations, for both types of models.
2.2.3. Ecological Explanations
The effect of each covariate of the generalized model was evaluated graphically, using
five equal intervals between the covariate bounds from Table 1, while using the mean values
of the other in two-dimensional charts [42]. Based on the chart form, ecological explanations
were attempted in order to investigate the processes behind tree height expression, along
with the other competition or productivity-related independent variables.
2.2.4. Analysis
For the statistical analysis, open-source R software was used [43]. For the mixedeffects model fitting along with the nlsList and the anova functions, the nlme library [17] was
implemented. The residual analysis was based on the lmfor package [14] and the graphical
illustrations on the ggplot2 package.
3. Results
The scatterplot of the height against the diameter is presented in Figure 2. During
model fitting, serious convergence problems were detected when all of the parameters
were assumed to be mixed in 3-parameters models. The mixed-effects parameters were
reduced but still convergence problems were detected in two sample plots, despite the
different combinations of the initial parameter estimates, which were further confirmed
Sustainability 2021, 13, x FOR PEER REVIEW
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19
the nlsList function. Hence, the 2-parameter models were preferred since they presented an
increased convergence ability.
Figure
Figure 2.
2. Scatterplot
Scatterplot of
of the
the diameter
diameter (dbh)
(dbh) against
against height
height (h),
(h), Pinus
Pinus nigra
nigra Arn.
Arn. National
National Forest
Forest Park
Park
of
Troodos,
Cyprus.
of Troodos, Cyprus.
The Naslund’s M1 model managed to converge when both parameters were assumed
to include a random part and it presented the best fittings compared to the other models
(Table 3).
Table 3. Fitting performance of the 2-parameter mixed models.
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The Naslund’s M1 model managed to converge when both parameters were assumed to include a random part and it presented the best fittings compared to the other
models (Table 3).
Table 3. Fitting performance of the 2-parameter mixed models.
Model
M1
M2
M3
M4
M5
0.086
0.721
1.204
0.723
Fixed parameters
β0
β1
2.611
0.173
45.697
29.613
64.814
0.025
Variance components
var(b0 )
var(b1 )
cov(b0 ,b1 )
σ2 (error variance)
0.2878
0.0002
−0.004
2.984
45.697
10.983
7.323
2.889
Goodness of fit
42.973
0.0007
−0.927
3.320
0.040
0.011
−0.021
3.905
0.0000
0.0007
0.0000
4.301
RMSE
FI
AIC
BIC
Bias
1.689
0.919
4310.30
4340.09
−0.052
1.866
0.901
4456.15
4485.94
−0.124
1.788
0.909
4419.05
4448.84
−0.119
1.937
0.893
4571.43
4601.22
−0.088
2.048
0.881
4642.47
4672.27
−0.100
The abbreviations of the fitting statistics are explained in Section 2.2.
The M1 model managed to predict almost 92% of the height variance according to
the reported Fitting Index and it presented the lowest error between the tested models
(Table 3). However, convergence problems emerged when other 2-parameter models were
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tested, such as Curtis’ [3] model. Figure 3 depicts the 95% confidence intervals on the
coefficients of the M1 model fitted to 28 sample plots.
Figure3.3.The
The95%
95%confidence
confidenceintervals
intervalson
onthe
thecoefficients
coefficients(β
(β0,1
0,1)) of
Figure
of the
the Näslund’s
Näslund’s model
model (M1)
(M1) fitted
fitted
tothe
thetotal
totalnumber
numberof
ofsample
sampleplots,
plots,Pinus
Pinusnigra
nigraArn.
Arn.National
NationalForest
ForestPark
ParkofofTroodos,
Troodos,Cyprus.
Cyprus.
to
Basedon
onthose
thosecoefficients,
coefficients,
the
M1
graphical
representation
(“spaghetti”
chart)
(Fig-4)
Based
the
M1
graphical
representation
(“spaghetti”
chart)
(Figure
ure 4) confirms
thefitting
good performance
fitting performance
of the
M1 mixed-effects
confirms
the good
of the M1
mixed-effects
model. model.
Figure 3. The 95% confidence intervals on the coefficients (β0,1) of the Näslund’s model (M1) fitted
to the total number of sample plots, Pinus nigra Arn. National Forest Park of Troodos, Cyprus.
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Based on those coefficients, the M1 graphical representation (“spaghetti” chart) (Figure 4) confirms the good fitting performance of the M1 mixed-effects model.
Figure4.4.“Spaghetti”
“Spaghetti”chart
chartofofthe
theNäslund’s
Näslund’smodel
model(M1)
(M1)
fitting
performance
against
observaFigure
fitting
performance
against
thethe
observations
tions
of
the
total
sample,
where
h
is
the
total
height
and
dbh
is
the
diameter
at
breast
height,
Arn.
of the total sample, where h is the total height and dbh is the diameter at breast height, Pinus nigraPinus
nigra Arn. National Forest Park of Troodos, Cyprus.
National Forest Park of Troodos, Cyprus.
Thetotal
totalsample
sampleand
andthe
thewithin-group
within-group
predictions
against
height–diameter
obThe
predictions
against
thethe
height–diameter
obser9 of 19
per
plot
(Figure
5)
further
configure
the
good
performance
of
the
suggested
vations per plot (Figure 5) further configure the good performance of the suggested model,
model,
the within-group
predictions
areagreement
in close agreement
with theconcentrations.
observed consince
thesince
within-group
predictions
are in close
with the observed
centrations.
Sustainability 2021, 13, x FOR PEER REVIEW
servations
Figure
predictions (fixed
(fixed lines),
lines),within-group
within-grouppredictions
predictions(plot
(plotlines)
lines)and
andheight–
height–
Figure 5.
5. Total sample predictions
diameterobservations
observationsper
perplot.
plot.The
Thenumbers
numberson
ontop
topcorrespond
correspondtotothe
theplot
plotnumber,
number,Pinus
Pinus
nigra
diameter
nigra
Arn.
Arn. National
Park
of Troodos,
Cyprus.
National
ForestForest
Park of
Troodos,
Cyprus.
In addition,
addition, the
the visual
visual inspection
inspection of
of the
the residuals
residuals revealed
revealed no
no obvious
obvious heteroscedastic
heteroscedastic
In
trends. However, when
when an
an exponential-function
exponential-function form
form was
was used
used for
for residual
residual regulation
regulation
trends.
(Table
was
improved
significantly
according
to the
anova
function
(Table44and
andFigure
Figure6),6),the
themodel
model
was
improved
significantly
according
to the
anova
func(p
< 0.01).
tion
(p < 0.01).
Hence, the final form of the basic model was as follows:
M + Equation (10)
Table 4. Comparison of variance functions with M1 mixed model.
Variance Function
Power
Independent Variable (x)
dbh
δ
−0.13118
AIC
4305.97
BIC
4340.72
log(L)
−2145.98
(12)
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Table 4. Comparison of variance functions with M1 mixed model.
Variance Function
Power
Constant-Power
Exponential
Independent Variable (x)
dbh
δ
AIC
BIC
log(L)
δ1
δ2
AIC
BIC
log(L)
δ
AIC
BIC
log(L)
−0.13118
4305.97
4340.72
−2145.98
0.00000
−0.13118
4314.29
4354.02
−2149.15
−0.00924
4288.07
4322.83
−2137.04
Where δ, δ1 and δ2 are the estimated parameters of the variance function, and AIC, BIC and log(L) are measures
of good fit, as explained in Section 2.2.
Figure 6. Standardized residual plot for the M1 mixed-effects model after weighting. The large
dots represent the mean values of the residuals according to ten classes. The thick and thin vertical
lines represent the 95% confidence interval of the class mean and of one observation (mean ± SD),
respectively. The red color indicates the lines that do not cross the base line at y = 0, Pinus nigra Arn.
National Forest Park of Troodos, Cyprus.
Hence, the final form of the basic model was as follows:
M1 + Equation (10)
(12)
The graphical interpretation of the Q–Q plot (Figure 7) revealed that the residuals did
not decline significantly from a straight line, and the normality assumption holds.
Figure 7. Normal Q–Q plot of the weighted M1 mixed-effects model, Pinus nigra Arn. National
Forest Park of Troodos, Cyprus.
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The final M1 model’s parameters and fitting statistics are presented in Table 5. Assuming independence between the observations, the error of the corresponding OLS model
was equal to 2.325 m, and while using only the fixed part of the model, the error was
slightly increased to 2.752 m, which was expected. However, the mixed-effects models
can be adjusted to each sample plot using a small sample of trees per plot, reducing the
prediction error to a large extent.
Table 5. Estimated parameters and fitting statistics of the basic mixed model (M1).
Mixed Model (M1)
Fixed Parameters (Standard Error)
β0
2.5612
(0.116)
FI
0.919
β1
0.1750
(0.003)
δ
−0.00924
RMSE
1.6905
Random Variance Components
var(b0 )
0.3070
var(b1 )
0.0002
Fit statistics
AIC
4288.07
cov(b0 ,b1 )
−0.0039
BIC
4322.83
The abbreviations of the fitting statistics are explained in Section 2.2.
σ2 (error variance)
4.985
Bias
−0.0592
The 95th quantile regression for nonlinear mixed-effects modeling of the maximum
crown width to estimate the ccf, provided the following equation:
cwmax = 0.5241dbh0.68971
(13)
Figure 8 presents the correlation between the model (M1) parameters and the basic
variables at the plot level.
The β0 parameter was negatively correlated with cbh (r = −0.539, p < 0.01), whereas the
β1 parameter was negatively correlated with hdom (r = −0.703, p < 0.01), ddom (r = −0.650,
p < 0.01) and qmd (r = −0.465, p < 0.01), and positively correlated with rsi (r = 0.385,
p < 0.05) and N/ha (r = 0.417, p < 0.05). However, the ccf did not correlate with any of the
two parameters.
For the development of the generalized form, the correlated variables were introduced
into the model and their effect was assessed using the significance of the associated parameter (approximate t-statistic). Furthermore, the number of the inherent variables was
intently tested to avoid any overfitting that could possibly occur. The hdom and the ddom
at the stand level, along with the cbh at the tree level, improved the model’s performance
significantly and they are proposed as additional covariates for Black pine height predictions. Despite the fact that the cbh at the plot level was correlated with β0 , the inclusion
of cbh at the tree level provided more accurate predictions and it was retained. The other
variables were not associated with a significant parameter within the model, and they were
not included in its final form. The relationships of the response variable and the covariates
in the final model are presented in Figures 9–11 and the form of the generalized model (G1)
is as follows:
h = 1.3 +
dbh2
(β0 + β1 dbh + β2 hdom + β3 ddom + β4 cbh)2
(14)
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cw
= 0.5241dbh
.
(13)
11 of 18
Figure 8 presents the correlation between the model (M1) parameters and the basic variables at the
plot level.
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Figure 8.
8. Correlations
Correlations between
) and
basic
variables
at at
thethe
plot
level.
Figure
between stand-level
stand-levelparameters
parameters(β
(β0,10,1
) and
basic
variables
plot
level.
The cbh is the mean crown base height, the hdom and the ddom are the dominant height and diameThe cbh is the mean crown base height, the hdom and the ddom are the dominant height and diameter,
ter, correspondingly, the rsi is the relative spacing index,
dbhthe qmd is the quadratic mean diameter
correspondingly, the rsih is
the
spacing index, the qmd is the quadratic mean diameter (14)
and
1.3relative
+ Pinus
and N/ha is the number =
of stems,
nigra
Arn.
National
Troodos, Cyprus.
(βnigra
+ βArn.
dbhNational
+ β h Forest
+Forest
β Park
d Park
βofcbh)
N/ha is the number of stems, Pinus
of+Troodos,
Cyprus.
The β0 parameter was negatively correlated with cbh (r = −0.539, p < 0.01), whereas
the β1 parameter was negatively correlated with hdom (r = −0.703, p < 0.01), ddom (r = −0.650,
p < 0.01) and qmd (r = −0.465, p < 0.01), and positively correlated with rsi (r = 0.385, p <
0.05) and N/ha (r = 0.417, p < 0.05). However, the ccf did not correlate with any of the two
parameters.
For the development of the generalized form, the correlated variables were introduced into the model and their effect was assessed using the significance of the associated
parameter (approximate t-statistic). Furthermore, the number of the inherent variables
was intently tested to avoid any overfitting that could possibly occur. The hdom and the
ddom at the stand level, along with the cbh at the tree level, improved the model’s performance significantly and they are proposed as additional covariates for Black pine height
predictions. Despite the fact that the cbh at the plot level was correlated with β0, the inclusion of cbh at the tree level provided more accurate predictions and it was retained. The
other variables were not associated with a significant parameter within the model, and
they were not included in its final form. The relationships of the response variable and the
Figure
9.
ofofthe
dominant
height
dom) against height (h), Pinus nigra Arn. National
Figure
9. Scatterplot
Scatterplot
the
dominant
height(h(h
) against9–11
height
Arn.
National
dom
covariates
inTroodos,
the final
model
are presented
in
Figures
and(h),
thePinus
formnigra
of the
generalized
Forest
Park
of
Cyprus.
Forest
Park
of
Troodos,
Cyprus.
model (G1) is as follows:
Sustainability 2021, 13, 5998
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Figure 9. Scatterplot of the dominant height (hdom) against height (h), Pinus nigra Arn. National
Figure
9. Scatterplot
of the
dominant height (hdom) against height (h), Pinus nigra Arn. National
Forest Park
of Troodos,
Cyprus.
Forest Park of Troodos, Cyprus.
Figure 10.
10. Scatterplot
of the
the dominant
dominant diameter
diameter (d
(ddom)) against
Figure
Scatterplot of
against height
height (h),
(h), Pinus
Pinus nigra
nigra Arn.
Arn. National
National
Figure
10. Scatterplot
of Cyprus.
the dominant diameter (ddom
dom) against height (h), Pinus nigra Arn. National
Forest
Park
of
Troodos,
Forest
Park
of
Troodos,
Cyprus.
Forest Park of Troodos, Cyprus.
Figure 11. Scatterplot of the crown base height (cbh) against height (h), Pinus nigra Arn. National
Figure
11.
of
the
Figure
11. Scatterplot
Scatterplot
ofCyprus.
thecrown
crownbase
baseheight
height(cbh)
(cbh)against
againstheight
height(h),
(h),Pinus
Pinusnigra
nigraArn.
Arn.National
National
Forest Park
of Troodos,
Forest
Park
of
Troodos,
Cyprus.
Forest Park of Troodos, Cyprus.
In the second phase, two of the five total parameters were expanded as mixed-effects
and the model was checked for heteroscedasticity by linking variance with one candidate
function. The model converged when β0 and β1 were assumed as mixed and the final form
was (GM1), as follows:
hi = 1.3 +
dbh2
(β0 + b0i + (β1 + b1i )dbh + β2 hdom + β3 ddom + β4 cbh)2
(15)
Table 6 presents the comparison of variance functions with the generalized mixed
model (GM1). The power function presented the best results, and it was selected to regulate
GM1 error’s variance.
Hence, the final model becomes the following:
GM1 + Equation (9)
(16)
The model’s parameter estimates, along with the fitting statistics, are shown in Table 7.
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Table 6. Comparison of variance functions with GM1 mixed model.
Independent Variable (x)
Variance Function
dbh
0.14083
3577.98
3627.64
−1778.99
0.00000
0.14083
3579.98
3634.60
−1778.99
0.00410
3582.25
3631.90
−1781.13
δ
AIC
BIC
log(L)
δ1
δ2
AIC
BIC
log(L)
δ
AIC
BIC
log(L)
Power
Constant-Power
Exponential
Where δ, δ1 and δ2 are the estimated parameters of the variance function.
Table 7. Estimated parameters and fitting statistics of the generalized mixed model (GM1).
Fixed Parameters (Standard Error)
β0
β1
4.737
(0.199)
0.210
(0.003)
FI
0.957
β2
−0.106
(0.016)
RMSE
1.228
Generalized Mixed Model (GM1)
Random Variance Components
β3
β4
var(b0 )
var(b1 )
cov(b0 ,b1 )
0.013
(0.006)
−0.152
(0.005)
0.2452
0.0002
−0.0065
Fit statistics
AIC
3577.98
BIC
3627.64
The abbreviations of the fitting statistics are explained in Section 2.2.
σ2 (error
variance)
0.624
Bias
−0.012
The residual and the Q–Q plot are presented in Figures 12 and 13, respectively. From
a visual inspection, no heteroscedastic trends were detected; neither was a significant
deviation from the straight line observed.
Figure 12. Standardized residual plot for the generalized mixed-effects model GM1 after weighting
with power function, Pinus nigra Arn. National Forest Park of Troodos, Cyprus.
Sustainability 2021, 13, 5998
Figure 12. Standardized residual plot for the generalized mixed-effects model GM1 after
weighting with power function, Pinus nigra Arn. National Forest Park of Troodos, Cyprus.
14 of 18
Figure
Figure 13.
13. Normal
Normal Q–Q
Q–Q plot
plot of
ofthe
theweighted
weightedGM1
GM1mixed-effects
mixed-effectsmodel,
model,Pinus
Pinusnigra
nigraArn.
Arn.National
National
Forest
Cyprus.
Forest Park
Park of
of Troodos,
Troodos, Cyprus.
Figures
a
Figures 14–16
14–16 depict
depict the
the separate
separatecontribution
contributionofofeach
eachcovariate
covariateofofthe
themodel
modelinin
total
height
allometry
of Black
pine
trees
in Cyprus.
The The
curves
werewere
placed
on a on
twoa total
height
allometry
of Black
pine
trees
in Cyprus.
curves
placed
a
dimensional
chartchart
for evaluation.
two-dimensional
for evaluation.
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Figure
effect
of of
thethe
dominant
height
(hdom
on )the
tree tree
Figure 14.
14. Graphical
Graphicalrepresentation
representationofofthe
the
effect
dominant
height
(h)dom
ontotal
the total
height
Cyprus.
height (h),
(h), Pinus
Pinus nigra
nigra Arn.
Arn. National
National Forest
Forest Park
Park of
of Troodos,
Troodos, Cyprus.
Figure 15.
15. Graphical
Graphicalrepresentation
representationofofthe
the
effect
crown
base
height
(cbh)
on the
Figure
effect
of of
thethe
crown
base
height
(cbh)
on the
totaltotal
tree tree
height
(h) Pinus
Pinus nigra
nigra Arn.
Arn. National
NationalForest
ForestPark
ParkofofTroodos,
Troodos,Cyprus:
Cyprus.
height (h)
The effect is valid when h ≥ cbh.
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Figure 15. Graphical representation of the effect of the crown base height (cbh) on the total tree
height (h) Pinus nigra Arn. National Forest Park of Troodos, Cyprus.
Figure 16.
16. Graphical
Graphical representation
representationof
ofthe
theeffect
effectof
ofthe
thedominant
dominantdiameter
diameter(d(d
) on
total
tree
Figure
dom
) on
thethe
total
tree
dom
height
(h),
Pinus
nigra
Arn.
National
Forest
Park
of
Troodos,
Cyprus.
height (h), Pinus nigra Arn. National Forest Park of Troodos, Cyprus.
4. Discussion
Discussion
4.
The Näslund
Näslundfunction,
function,which
whichhas
has
been
used
several
other
species
its
The
been
used
forfor
several
other
species
due due
to itstoadeadequate
flexibility
[1],
presented
good
fitting
ability
for
Black
pine
height
predictions
too.
quate flexibility [1], presented good fitting ability for Black pine height predictions too.
Mehtätalo et
et al.
al. [14]
[14] suggested
suggested the
the Näslund
Näslund function
function as
as the
the best
best among
among other
other 2-parameter
2-parameter
Mehtätalo
models
for
height–diameter
mixed-effects
modeling,
while
Sharma
et
al.
used
it
models for height–diameter mixed-effects modeling, while Sharma et al. [44][44]
used
it for
for
height
predictions
of
conifer
and
broadleaved
species
in
the
central
Czech
Republic.
height predictions of conifer and broadleaved species in the central Czech Republic.
Schmidt et
et al.
al. [45]
[45] also
also used
used the
the linear
linear transformation
transformation of
of M1
M1 model
model for
for height
height predictions
predictions
Schmidt
of Scots pine (Pinus sylvestris) in Estonia. Due to the shade-intolerance and the even-aged
of Scots pine (Pinus sylvestris) in Estonia. Due to the shade-intolerance and the even-aged
structure of the Black pine forest stands, special conditions have been generated in the
structure of the Black pine forest stands, special conditions have been generated in the
Troodos Mountains that reflect the absence of small diameter trees in the understory of
Troodos Mountains that reflect the absence of small diameter trees in the understory of
some of the stands. This actually prevented some models from converging, while the
some of the stands. This actually prevented some models from converging, while the
Näslund function managed to converge mainly due to its parsimony and ability to be
Näslund function managed to converge mainly due to its parsimony and ability to be
linearized from a mathematical perspective. Overall, the suggested generalized model
linearized from a mathematical perspective. Overall, the suggested generalized model
(GM1) explained almost 96% of the height variance (Table 7). This can be attributed to the
(GM1) explained almost 96% of the height variance (Table 7). This can be attributed to the
flexibility of the basic model, the introduced additional covariates and the random effect
flexibility of the basic model, the introduced additional covariates and the random effect
of the two model parameters. The remaining 4% of unexplained height variance can be
of the two model parameters. The remaining 4% of unexplained height variance can be
attributed to variables or factors that are not included in the proposed model. Further
attributed to variables or factors that are not included in the proposed model. Further
improvements using additional covariates may lead to overfitting conditions due to an
increased complexity. Therefore, the unexplained portion was judged to be quite small and
the height allometry well captured by the GM1 generalized mixed-effects model.
From a modeling process perspective, the additional covariates are critical as far as
the model’s biological validity is concerned. In a relevant study, Sharma and Zhang [46]
used the stand basal area (m2 ·ha−1 ) and the stand density (n·ha−1 ) to explain the height
allometry of both Jack pine (Pinus banksiana) and Black spruce (Picea mariana) in Canada.
Gómez-García et al. [47] incorporated the dominant height, the number of stems and
the dominant diameter in a mixed-effect model to develop a generalized model for the
height estimation of maritime pine (Pinus maritima) stands in Portugal, while Sharma and
Breidenbach [1] introduced dominant height and diameter into Näslund’s function to
model the height variable of Scots pine (Pinus sylvestris), Norway spruce (Picea abies) and
downy birch (Betula pubescens) in Norway. In general terms, the proposed covariates in each
of the forming modeling procedures were related to the site productivity, the competition
status and the development stage of forest stands and explained effectively the height
variance for different tree species, which is in agreement with the current study.
The Site Index (SI) reflects the combined effect of physical and ecological factors,
which are expressed though the site’s ability to support tree growth [48]. It is closely linked
with the hdom , since it is estimated from the mean height of the largest trees at a reference
Sustainability 2021, 13, 5998
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age [49]. The effect of hdom on the height expression of Black pine trees suggests that the
height of trees with an equal height to crown base height, grown in similar competition
conditions (dominant diameter) is larger in sites where the height of the dominant trees
is higher (Figure 14). From an ecological perspective, this was expected, since dominant
height also reflects the development stage of the stand (Figure 9). In this sense, the highest
dominant height implies higher individual heights due to the older stand age [50]. On
the other hand, a higher dominant height may be attributed to better site quality and,
consequently, better growth conditions. The overall positive effect of hdom is consistent
with the findings from other authors and it has been included in the height modeling
procedure [30,51–53]. From an operational perspective, the inclusion of the hdom variable
as a covariate for the development of generalized models is advantageous, because it can
be easily estimated in the field by measuring only a few trees per stand [54].
The ddom has the lowest effect on height allometry compared to the other two covariates (Figures 10 and 16). Moreover, a negative impact on tree heights can be observed,
which was expected, implying that the largest dominant trees in stands of the same development stage lead to lower individual tree heights. This outcome complies with the findings
of Castredo-Dorado et al. [52] and Castaño-Santamaría et al. [55], who incorporated the
effect of ddom into their proposed generalized models. This impact may be attributed to
the effect of the competition status within the stand reflected through the ddom . Under
this hypothesis, sparse stands lead to the thickest dominant trees in sites where lower tree
heights occur due to lower competition levels within the stand. The competition level and
the effect on tree heights have been already well established in forest science [52].
The cbh at the tree level has the largest impact on Black pine tree height (Figure 15).
The chart interpretation leads to the hypothesis that trees of same stage development,
which grow under similar competition conditions, present a positive relationship between
cbh and total height. The isolated positive relationship between the two variables is further
verified in Figure 11. The ecological explanation behind the observed impact may be
attributed to the self-pruning capacity of the specific species and the light availability at the
crown base status within each stand. Dense stands favor height development by creating
shade conditions at the lower part of the tree crowns where self-pruning may occur, thus
resulting in increased height to live crowns. A similar hypothesis has been also expressed
by Sharma et al. [42], who introduced the competitive stress caused by the crowding of the
trees in order to explain the cbh allometry. Overall, the increased crowding of trees leads to
taller trees, thinner stem diameters and larger cbhs due to the crown recession [56].
5. Conclusions
A generalized mixed-effects model for height predictions of Black pine trees has
been developed. The model explains almost 96% of height variance, incorporating site,
competition and tree-level variables. Through the mixed-effects modeling approach, it was
possible to explain complex ecological processes behind the Black pine tree allometry and
express those within a mathematical formulation. For operational and marginal predictions
of the standing woody volume in the frame of sustainable forest management, the fixed
part of the M1 model is suggested for use, since it provides unbiased parameter estimates,
despite the slightly increased prediction error, when compared to the corresponding OLS
model. The generalized model decreased errors significantly and it is also recommended
for operational sustainable forest management purposes, especially when data for crown
base height at tree level are available for processing. The future prospects of the current
research will include the validation of the proposed models in different regions through
mixed-effect modeling, using a small sample of trees to localize the random part.
Author Contributions: Conceptualization, D.I.R.; methodology, D.I.R.; validation, D.I.R.; formal
analysis, D.I.R., V.K., C.S. and A.K.; investigation, D.I.R., V.K., C.S. and A.K.; resources, D.I.R. and
V.K.; data curation, N.O.; writing—original draft preparation, D.I.R.; writing—review and editing,
V.K. and A.K.; visualization, C.S. All authors have read the term explanation and agreed to the
published version of the manuscript.
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Funding: This research received no external funding.
Institutional Review Board Statement: Not applicable.
Informed Consent Statement: Not applicable.
Data Availability Statement: The data are available on reasonable request.
Acknowledgments: We thank greatly the Department of Forests of the Cyprus Ministry of Agriculture, Rural Development and Environment for providing the spatial information of the Black pine
distribution in the Troodos Mountains. We also thank greatly the three anonymous reviewers for
providing valuable comments on improving the quality of our paper.
Conflicts of Interest: The authors declare no conflict of interest.
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