Allometric Model of Crown Length for Pinus sylvestris L. Stands in South-Western Poland

Abstract

The growth of a tree depends on the size, shape, and functioning of the crown. The length of the crown is a somewhat subjective value because the base of the crown is often difficult to determine. The aim of this study was to develop an allometric model to calculate the crown length of Pinus sylvestris L., which might serve as an alternative to the current equations used especially for stands of variable density. The model used three predictive variables, i.e., diameter at breast height, tree height, and density. The developed crown length model showed high compatibility with empirical data within the studied stands differing in diameter at breast height, height, age, biosocial position, and, above all, density (SD = 1.786). The correlation coefficient between the empirical crown length for the stand ($H_{emp}^{*}$) and the calculated model $\left(H_{cal}^{*}\right)$ was r = 0.974, with a discrepancy of (±) 3.17%. The derived crown length model can be one of the components used to estimate the mass of needles or leaf area index (LAI) and, consequently, the amount of transpiration or the amount of carbon dioxide bound, which is crucial in the context of climate change.

1. Introduction

From an economic and environmental point of view, Scots pine (Pinus sylvestris L.) is one of the most important forest tree species, and it is the second largest in terms of its range among all coniferous species of the northern hemisphere [1,2]. In Poland, it is the dominant species that currently makes up about 58% of the forest species composition. This is due to the habitats typical for pine, i.e., coniferous and mixed coniferous forests, which cover about 50% of the forest area [3].

The size, structure, and shape of tree crowns determine the scope and efficiency of physiological processes, mainly photosynthesis, respiration, and transpiration, determining the growth and development of the tree. Estimates of crown size are often used to infer the tree’s vigour [4,5,6,7,8], basal area increment [8,9], production efficiency [10,11], and the quality and value of the wood [12,13,14,15,16]. In addition, knowledge of the size and shape of crowns is essential to predict the mechanical stability of trees [17] and their susceptibility to wind and snow damage [18,19,20,21], as well as vulnerability to fire [22,23,24]. It also helps to visualise the structure of trees and stands [18]. Furthermore, the crown length is used in allometric biomass models of biomass components, such as live and dead branches, stem bark, and leaves [25].

Burger [26,27,28,29,30,31] carried out extensive research on the structure and shape of tree crowns of various species. His research contributed to the development of crown models of such species as spruce, pine, or beech. In Poland, research on the relationship between crown size and tree growth in pine stands was carried out by Lemke [32,33,34]. Many works have focused on the relationship between the size of the crown and other features of the tree [35,36,37,38]. Diameter at breast height (DBH) is the most commonly used independent variable in crown width modelling, since the relationship between DBH and the crown width has been well established in the literature [39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54].

The length of the tree crown is of great importance in the methodology of breeding research [8,55,56,57,58]. This is because both the growth of the tree and the quality of the trunk depend on the length of its crown [59,60]. The length of the crown is difficult to determine if its shape is irregular. In forestry practice, it is measured from the top to the lowest green branch, maintaining continuity with the crown of the tree. Grochowski [61] considered the length of the crown to be the length from the top of the stem to the base of the first live branch, “maintaining continuity with the higher part of the crown”. If the crown is regular, that is, properly built, then its length is the distance from the treetop to the lowest base of the whorl of live branches.

Direct crown length measurements can be difficult to perform, especially in dense stands. Therefore, research works have begun to focus on the development of models for the crown length or relative crown length (i.e., the ratio of crown length to tree height). Numerous authors [62,63] have used simple geometric shapes for modelling. Other scientists began to use more flexible models to represent crown shapes [64,65,66,67,68]. Baldwin and Peterson [69] developed a series of equations to predict the outer and inner crown profile of Pinus taeda L., whereas Doruska and Mays [70], in an attempt to improve the crown profile estimate for this species, used nonparametric regression with only one predictive variable. The modelling was based on relationships between the tree and stand features [8,71,72,73]. Crown size predictions are usually based on allometric relationships with easy-to-measure tree trunk dimensions, such as tree height (H) and diameter at breast height (DBH). The use of these two biometric parameters for modelling seems justified because the size of the foliage determines the growth of trees in terms of their height and thickness [8,60,69,74,75,76]. In the case of stands growing at a low density (where there is no tree crown closure) or overpopulated stands, the use of these two parameters in allometric equations (H; DBH) might be insufficient to accurately determine the size of the crown [35,76,77]. Such equations may tend to overestimate the crown size in compact stands and underestimate it in open stands [78]. In such cases, it seems reasonable to add parameters related to competition, for example, taking density into account [35,41,54,79,80,81].

The aim of our research was to develop an equation to calculate the crown length ($H_{cal}^{*}$), which could serve as an alternative to current allometric equations, which are used especially in stands of varying density. It was hypothesised that of the three parameters, i.e., height, breast height, and density, the latter would have the greatest influence on crown length in pine stands. The following principal criteria were used to develop the model: (i) the model should account for the effect of density on the crown size; (ii) easy-to-measure biometric parameters of the tree, i.e., the height ($H$) and the diameter at breast height ($d_{1.3}$—DBH) should be used to develop the equation; (iii) the equation should be capable of providing an accurate calculation of the crown length for Scots pine stands of the second age class (21–40 years old) growing in coniferous habitats; and (iv) the equation should include trees with different biosocial positions in line with Kraft’s classification.

2. Materials and Methods

2.1. Description of the Study Site

The research was conducted in single species stands of Scots pine (Pinus sylvestris L.) in the south-western region of Poland. The research area was located between 18°35′ E and 19°20′ E, and 51°00′ N and 51°13′ N. The mean total of precipitation amounted to 641 mm·year⁻¹, with the heaviest precipitation recorded in July (118.5 mm) and the lowest in February (31.7 mm). The average annual temperature was 8.1 °C. July was the warmest (19.2 °C) and January was the coolest (−1.6 °C) month. The growing season lasted for ~222 days [82]. In the managed forests, 20 experimental plots of 0.25 ha (50 × 50 m) were designated. The age of the pine stands ranged from 30 to 38 years (Table 1). In the examined pine stands, there was a lot of dead wood due to density because no intermediate felling was made in them. The stands grew on podzols and brunic arenosols in the habitat conditions typical for pine, i.e., fresh or mixed fresh forest.

2.2. Biometric Measurements

On the designated plots, the diameter at breast height (DBH =$d_{1.3}$) was measured for all 14,935 trees. The DBH of the tree stands was in the range of 9 to 30 cm. This parameter was measured with a metal caliper Mantax Blue (Haglöf, Långsele, Västernorrland, Sweden) in two directions, N-S and W-E, with an accuracy of 0.1 cm, and the arithmetic mean of these measurements was assumed as the DBH. Based on the DBH measurement of all trees on each plot, 15 sample trees were randomly selected, taking into account dominant, codominant, intermediate, and suppressed trees. The selection was made using the English method developed by Humme felling every k-th tree based on the formula [83]:

$$k=\frac{N}{n}$$

(1)

where: $N$, total number of trees in a research area; $n$, the assumed number of sample trees.

During this felling, sample trees with different DBH, height, and crown length were obtained.

The selected sample trees were felled in March before the start of the growing season. A total of 300 trees were felled. After felling, their length was measured with a metal tape to the nearest 0.01 m and taken as the tree height ($H$) and crown length ($H_{emp}^{*}$). In this study, the following method was used to measure the crown length of the trees. A correctly formed whorl of a Scots pine has six branches. In this case, the crown is regular (Figure 1), and its length is defined as the distance from the treetop to the base of the whorl. If in the lowest whorl, one branch (of the size corresponding to the other branches) is dead, the length of the crown is the distance from the treetop to the base of the “incomplete” whorl, minus 1/6 of the distance between this base and the base of the nearest whole whorl (Figure 1). From this assumption, it follows that the crown length is reduced by 1/6 of the annual growth of the stem. In other words, if there are 2 dead branches in the last incomplete whorl, the measurement is shortened by 2/6 of the annual growth.

The above relationship is described by equation [35]:

$$H^{*}=\left(H-h\right)-\frac{n}{6\Delta }$$

(2)

where: $H^{*}$, crown length; H − h, distance from the treetop to the base of the whorl with withered (missing) branches; n, the number of withered (missing) branches in the whorl.

For each plot, the measured or calculated statistics for the stand were as follows: $d_{1.3}$, diameter at breast height (1.3 m above ground level); $H$, trees’ height; $H_{emp}^{*}$, empirical crown length $G_{1.3}$, stand basal area; $t$, the age of trees; $N$, the number of trees per hectare; $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])). The biometric description of the sites is presented in Table 1.

Table 1. Characteristics of surveyed Scots pine stands.

Variable	Mean	Minimum	Maximum	Median	Standard Deviation
$H$ (m)	13.9	7.0	21.1	13.5	3.20
$H_{emp}^{*}$ (m)	5.1	2.4	9.4	4.9	1.24
$d_{1.3}$ (cm)	13.0	6.0	24.0	13.0	3.37
$d_0$ (cm)	14.7	7.7	25.7	14.7	3.37
$N$ (trees ha−1)	2987	492	8340	3174	1786.4
$G_{1.3}$ (m2 ha−1)	24.0	11.7	32.6	23.9	4.85
$z$ (-)	1.07	0.31	1.86	1.14	0.40
$t$ (year)	33	30	38	33	2.39

$H$, trees height; $H_{emp}^{*}$, empirical crown length; $d_{1.3}$, diameter at breast height (1.3 m above ground level); $d_0$, the diameter of the trees at the ground level ($d_0=d_{1.3}+1.7 \mathrm{cm}$); $N$, the number of trees per hectare; $G_{1.3}$, stand basal area; $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])); $t$, age of trees.

Table 1. Characteristics of surveyed Scots pine stands.

Variable	Mean	Minimum	Maximum	Median	Standard Deviation
$H$ (m)	13.9	7.0	21.1	13.5	3.20
$H_{emp}^{*}$ (m)	5.1	2.4	9.4	4.9	1.24
$d_{1.3}$ (cm)	13.0	6.0	24.0	13.0	3.37
$d_0$ (cm)	14.7	7.7	25.7	14.7	3.37
$N$ (trees ha−1)	2987	492	8340	3174	1786.4
$G_{1.3}$ (m2 ha−1)	24.0	11.7	32.6	23.9	4.85
$z$ (-)	1.07	0.31	1.86	1.14	0.40
$t$ (year)	33	30	38	33	2.39

$H$, trees height; $H_{emp}^{*}$, empirical crown length; $d_{1.3}$, diameter at breast height (1.3 m above ground level); $d_0$, the diameter of the trees at the ground level ($d_0=d_{1.3}+1.7 \mathrm{cm}$); $N$, the number of trees per hectare; $G_{1.3}$, stand basal area; $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])); $t$, age of trees.

2.3. Dependency Detection

We assume that there is an allometric relationship between the size of one organ of an individual ($y)$ and the size of another part of their body ($x)$:

$$y=C{x}^{\alpha }$$

(3)

Thus, the allometric relationship between the length of the tree crown $H^{*}$ and some other measure of its size $x$ can be expressed as follows:

$$H^{*}=C{x}^{\alpha }$$

(4)

We do not know the value of the factor of proportionality $C$, the exponent $\alpha$, or exactly what features fit into the measure $x$. Previous studies [8,60,69,74] have indicated that the measure $x$ consists of the thickness of the tree and its height, as well as the stand density [11,85]. Therefore, we are left with a task to develop an allometric model that would allow us to calculate the length of the crown based on the collected empirical data.

3. Results

3.1. Analysis of Data for Building the Crown Length Model

It was assumed that the length of the tree crown in a single species stand (Pinus sylvestris L.) would depend mainly on stand density. The relationships between DBH, height, density, and crown length for 300 sample trees are presented in

Table 2. Pearson correlation matrix between crown length and biometric features and density of 300 sample trees. Statistically significant results for p < 0.05.

		${\mathit{H}}_{\mathit{e}\mathit{m}\mathit{p}}^{*}$	${\mathit{d}}_{1.3}$	$\mathit{H}$	z
		(m)	(cm)	(m)	(-)
$d_{1.3}$ (cm)	Pearson’s r	0.7862	–
	p-value	<0.0001	–
$H$(m)	Pearson’s r	0.7216	0.8536	–
	p-value	<0.0001	<0.001	–
$z$ (-)	Pearson’s r	−0.8728	−0.4735	−0.3751	–
	p-value	<0.0001	<0.0001	<0.0001	–

$H_{emp}^{*}$, empirical crowns length; $d_{1.3}$, diameter at breast height (1.3 m above ground level); $H$, trees height; $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])).

.

The correlation coefficients ranged from −0.375 to −0.873, and they were clearly higher than the critical values for the assumed significance level of 0.05. The highest correlation coefficient was found for $z$ and $H_{emp}^{*}$ (r = −0.873; p < 0.0001). In this case, a linear negative correlation was determined. The higher the density of the stand, the shorter the crowns were (Figure 2a). In the case of a positive correlation between $H_{emp}^{*}$ and d_1.3, the coefficient was r = 0.786; p < 0.0001 (Figure 2b). A slightly weaker relationship was found between $H_{emp}^{*}$ and $H$ (r = 0.722; p < 0.0001) (Figure 2c).

The density of the studied pine stands differed from 492 to 8340 trees per hectare (

Table 3. Summary statistics of the final dataset used to fit the crown length equation (n = 300 trees measured in 20 plots).

Plot Number	${\overline{\mathit{d}}}_{1.3}$	$\overline{\mathit{H}}$	${\overline{\mathit{H}}}_{\mathit{e}\mathit{m}\mathit{p}}^{*}$	$\mathit{z}$	$\mathit{N}$	$\mathit{t}$
	(cm)	(m)	(m)	(-)	(Trees/ha)	(Year)
1	10.5	9.70	4.51	1.12	4180	30
2	12.2	13.41	4.84	1.23	3208	30
3	12.3	12.64	4.46	1.15	3192	31
4	9.3	12.52	4.29	1.25	3300	33
5	10,9	12.34	4.42	1.40	3628	34
6	12.5	13.41	4.88	0.97	2312	35
7	12.5	13.43	4.71	1.36	3544	32
8	12.7	13.95	5.35	0.93	2512	30
9	10.7	12.06	4.33	1.86	4928	35
10	15.6	16.97	5.35	1.27	2268	38
11	18.5	18.96	6.46	0.61	860	38
12	18.3	19.18	6.99	0.31	492	33
13	14.5	16.38	5.86	0.54	1188	33
14	16.0	17.60	5.71	0.63	1252	32
15	15.7	15.50	5.86	0.48	1092	33
16	8.7	8.36	3.84	1.04	5080	30
17	8.9	8.61	3.98	1.83	8340	31
18	16.1	18.27	6.24	0.84	1468	35
19	13.1	13.12	4.44	1.30	3156	35
20	11.7	12.94	4.75	1.23	3740	32
Mean	13.0	13.97	5.06	1.7	2987	-

${\overline{d}}_{1.3}$, average diameter at breast height (1.3 m above ground level); $\overline{H}$, average trees height; ${\overline{H}}_{emp}^{*}$, average empirical crowns length; $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])); $t$, age of trees; $N$, the number of trees per hectare.

). Biometric parameters of the studied stands (${\overline{d}}_{1.3}; \overline{H}; {\overline{H}}_{emp}^{*})$ calculated for 15 sample trees for each experimental plot are summarised in Table 3. Empirical data showed that at the density index of $z$ = 1.83, the average crown length was the shortest and equalled ${\overline{H}}_{emp}^{*}$= 3.98 m, while at the lowest density of $z$ = 0.31, the crown was the longest, amounting to ${\overline{H}}_{emp}^{*}$= 6.99 m. The greater the space between the trees, the longer the crown was. For the final dataset, the summary statistics were used to fit the crown length equation (Table 3).

3.2. Crown Length Modelling

Three predictive variables, i.e., DBH, height, and density index ($d_{1.3};H;z)$, were used to build the crown length model. The density index was defined as the ratio of the actual number of trees per hectare (which we know only at the time of measurement) to the expected number of trees in a given age and stand quality class (value read from Tables of Stands Volume and Increment [84]). It seems that the introduction of this predictor into the model was essential, as the collected empirical material exhibited a large variation in the number of trees on individual experimental plots.

The analysis of the data in Table 3 confirmed the emerging assumption about the size of measure x in Equation (4):

$$x~H{d}_{1.3}\frac{1}{\sqrt{z}}$$

(5)

where: $H$, trees height (m); $d_{1.3}$, DBH (cm); $z$, density index.

Since the diameter at breast heights ($d_{1.3}$) is a technical value, it can be replaced by the diameter of the tree at ground level ($d_0$) [85]:

$$d_0=d_{1.3}+1.7\left[\mathrm{cm}\right]$$

(6)

A graphical image of the crown length as a function of density, DBH, and height is illustrated in Figure 3.

The points representing the crown length were arranged around the line according to the equation:

$${\overline{H}}_{cal}^{*}~C{\left(\overline{H}{\overline{d}}_0\frac{1}{\sqrt{z}}\right)}^{\alpha }$$

(7)

where: ${\overline{H}}_{cal}^{*}$, the average calculated crown length (m); $\overline{H}$, average trees’ height (m); ${\overline{d}}_0$, average diameter of the base of the tree (cm); $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])); $C$, factor of proportionality—dimensionless number (-); $0<\alpha <1$.

In the first attempt to calculate the value of the index $C$ and the exponent $\alpha$, four stands were selected, i.e., the stand with the highest density index (plot No. 9), maximum height (plot No. 12), maximum DBH (plot No. 11), and the shortest crown (plot No. 17). The parameters of the selected stands are listed in

Table 4. Biometric parameters of four selected stands for determining the factor of proportionality $C$ and exponent $\alpha$ using the Smolik algorithm [86].

Plot Number	$\overline{\mathit{H}}$	${\overline{\mathit{d}}}_{\mathit{o}}$	$\mathit{z}$	${\overline{\mathit{H}}}_{\mathit{e}\mathit{m}\mathit{p}}^{*}$
	(m)	(cm)	(-)	(m)
17	8.61	10.06	1.83	3.98
9	12.06	12.40	1.86	4.33
11	18.96	20.20	0.61	6.46
12	19.18	20.00	0.31	6.99

$\overline{H}$, average trees height; ${\overline{d}}_0$, average diameter of the base of the tree, ${\overline{H}}_{emp}^{*}$, average empirical crowns length; $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])).

.

To calculate the values $C$ and $\alpha$, the Smolik algorithm was used [86]:

$$\alpha =\frac{{{\displaystyle \sum }}_{i=1}^n{y}_i^2{{\displaystyle \sum }}_{i=1}^n{y}_i^2\ln y_i\ln x_i-{{\displaystyle \sum }}_{i=1}^n{y}_i^2\ln x_i{{\displaystyle \sum }}_{i=1}^n{y}_i^2\ln y_i}{{{\displaystyle \sum }}_{i=1}^n{y}_i^2{{\displaystyle \sum }}_{i=1}^n{y}_i^2{\left(\ln x_i\right)}^2-{\left({{\displaystyle \sum }}_{i=1}^n{y}_i^2\ln x_i\right)}^2}$$

(8)

$$C=\exp \left(\frac{{{\displaystyle \sum }}_{i=1}^n{y}_i^2\ln y_i-\alpha {{\displaystyle \sum }}_{i=1}^n{y}_i^2\ln x_i}{{{\displaystyle \sum }}_{i=1}^n{y}_i^2}\right)$$

(9)

where: $y_i$; $x_i$—are the measured values; $y_i=H_{emp}^{*}$, empirical crown lengths (m); $x_i=\frac{\overline{H}{\overline{d}}_0}{\sqrt{z}}$ is the quotient of the products of the average height and the average diameter of the tree at ground level in the numerator to the root of the density index in the denominator.

The values obtained from the four sample plots, $\alpha$= 0.2505 and $C$ = 1.3621, were approximate values. Thus, the equation $H_{cal}^{*}$ can be written as follows:

$$H_{cal}^{*}=C{\left(H{d}_0\frac{1}{\sqrt{z}}\right)}^{0.25}$$

(10)

$$H_{cal}^{{*}^2}=\left(C^2\right)\sqrt{H{d}_0\frac{1}{\sqrt{z}}}$$

(11)

This means that the square of the crown length is directly proportional to the geometric mean height and diameter of the tree at ground level. Such a relationship is logical and seems to reflect the actual situation, where $H_{cal}^{{*}^2}$ is equivalent to the leaf area index, while the geometric mean height and diameter of the tree at ground level express the effect of the assimilation.

Therefore, adopting $\alpha =0.25$, the factor of proportionality $C$ was calculated as an average for the twenty studied stands based on the following Equation (12):

$$C=\frac{{\overline{H}}_{emp}^{*}}{{\left(\frac{\overline{H}{\overline{d}}_0}{\sqrt{z}}\right)}^{0.25}}$$

(12)

This way, the mean value of the factor of proportionality $C=1.33$ for the stands was obtained (

Table 5. Calculated value of the factor of proportionality ($C$) for all empirical material and determination of the efficiency of the developed allometric crown length model (${\overline{H}}_{cal}^{*}$).

Plot Number	$\mathit{z}$	${\overline{\mathit{d}}}_{\mathit{o}}$	$\overline{\mathit{H}}$	${\overline{\mathit{H}}}_{\mathit{e}\mathit{m}\mathit{p}}^{*}$	$\mathit{C}$	${\overline{\mathit{H}}}_{\mathit{c}\mathit{a}\mathit{l}}^{*}$	Divergence
	(-)	(cm)	(m)	(m)	(-)	(m)	(%)
1	1.12	12.2	9.70	4.51	1.39	4.3	4.10
2	1.23	13.9	13.41	4.84	1.34	4.8	1.06
3	1.15	14.0	12.64	4.46	1.24	4.8	6.88
4	1.25	11.0	12.52	4.29	1.29	4.4	3.28
5	1.40	12.6	12.34	4.42	1.31	4.5	1.88
6	0.97	14.2	13.41	4.88	1.31	5.0	1.63
7	1.36	14.2	13.43	4.71	1.32	4.8	0.98
8	0.93	14.4	13.95	5.35	1.41	5.1	5.56
9	1.86	12.4	12.06	4.33	1.34	4.3	0.60
10	1.27	17.3	16.97	5.35	1.33	5.3	0.13
11	0.61	20.2	18.96	6.46	1.37	6.3	3.12
12	0.31	20.0	19.18	6.99	1.36	6.8	2.52
13	0.54	16.2	16.38	5.86	1.34	5.8	1.06
14	0.63	17.7	17.60	5.71	1.28	5.9	3.67
15	0.48	17.4	15.50	5.86	1.32	5.9	0.81
16	1.04	10.4	8.36	3.84	1.26	4.0	5.24
17	1.83	10.6	8.61	3.98	1.39	3.8	4.23
18	0.84	17.8	18.27	6.24	1.44	5.8	7.49
19	1.30	14.8	13.12	4.44	1.23	4.8	8.21
20	1.23	12.2	12.94	4.75	1.34	4.7	0.99
Mean	1.7	14.7	13.97	5.06	1.33	5.1	3.17

$t$, age of trees; $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])); ${\overline{d}}_o$, average diameter of the trees at ground level; $\overline{H}$, average trees height; ${\overline{H}}_{emp}^{*}$, average empirical crowns length; ${\overline{H}}_{cal}^{*}$, average calculated crowns length; $C$, factor of proportionality.

).

Thus, equation $H_{cal}^{*}$ takes the following form:

$$H_{cal}^{*}=1.33{\left(H{d}_0\frac{1}{\sqrt{z}}\right)}^{0.25}$$

(13)

where: $H_{cal}^{*}$, calculated crowns length (m); H, trees’ height (m); $d_0$, diameter of the base of the tree (cm); $z$, density index (the actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [84])); factor of proportionality $C=1.33$.

The value of the factor of proportionality $C=1.33$ was identical for ${\pi }^{0.25 }=1.33$.

Finally, the model (13) can be represented as:

$$H_{cal}^{*}={\left(\pi d_{0}H\frac{1}{\sqrt{z}}\right)}^{0.25}$$

(14)

where: $\pi d_{0}H$ denotes the surface of a cylinder with diameter $d_0$ and height $H$.

In stands, where $z=1$, i.e., and the density is optimal for the growth and development of trees, Equation (14) has the following form:

$$H_{cal}^{*}={\left(\pi d_{0}H\right)}^{0.25}$$

(15)

The developed allometric crown length model (13) was used to calculate ${\overline{H}}_{cal}^{*}$ for the 20 analysed stands. The correlation coefficient between the empirical crown length for the stand ($H_{emp}^{*}$) and the calculated model $\left(H_{cal}^{*}\right)$ was r = 0.974 (p < 0.0001), with a discrepancy of (±) 3.17% (Table 5). The dispersion of empirical values $H_{emp}^{*}$ and the calculated ones $H_{cal}^{*}$ for the stands and model trees are presented in Figure 4a,b.

4. Discussion

In many respects, the length of the crown is an interesting biophysical property. However, in forestry surveys, it is rarely measured at both the tree and stand level. This is justified by the fact that the gain from having empirical information is considered lower than the cost of measuring the length of the crown [38]. Therefore, equations that accurately represent actual crown lengths are still being sought. In principle, crown length can be predicted on the basis of standard variables measured in forest management [38,87,88]. However, it is believed that it is very difficult to accurately calculate the length of the crown using basic measurements of trees, i.e., DBH, height, basal area, and stand age, without calibration at the stand level [38]. In our study, the crown length of the sample trees showed a strong correlation with both the DBH (r = 0.786; p < 0.0001) and height (r = 0.722 p < 0.0001) of the trees. Confirmation of such a relationship can be found in the research by Żybura [60], who, on the basis of these features, built tables of stand crown lengths for Pinus sylvestris. The mean error of these tables when determining the absolute crown length was 5.6%. Turski et al. [89] demonstrated that the relationship between the crown length and tree height was weaker than that between the crown length and DBH. However, we must accept the fact that there is much greater variation in crown length between different Pinus sylvestris stands than within a single stand [90]. In our study, the coefficients of variation in the crown length within one stand ranged from 10.8% to 23.3%. However, for all 20 examined stands, the coefficient of variation was higher and amounted to 24.5%. Such a large variation in crown length was influenced by, among others, the density of the examined tree stands. The correlation coefficient between these features was r = −0.873; p < 0.0001 (Table 2). Confirmation of this relationship can be found in the research of other authors [54,78,79,80,81].

Simple crown equations that restrict the predictor variables to one or two measurements of tree size are mostly useful in situations in which the differences in densities from stand to stand are limited while the stand structure is simple [76]. Gill et al. [91] stated that their use of only DBH to estimate the crown radius was likely helped by the fact that their data had been collected primarily from managed forests that had been treated to maintain a low density. Among several tested variables relating to density, studies related to crown allometry identified the basal area as the parameter that best described the effect of density on crown length [71,77].

The proposed crown length model $H_{cal}^{*}$ requires the measurement of $d$, $H$, and $z$ as input variables. The model shows high compatibility with empirical data within the studied stands (r = 0.974) (Figure 4), which differed in their DBH, height, age, biosocial position, and, above all, density (SD = 1786) (Table 1). The high efficiency of the model indicates the accuracy of its design and the selection of the parameters $C$ and $\alpha .$ These discrepancies come from measurement inaccuracies and incomplete representativeness of the model trees, which could be indicated by a lower coefficient of correlation (r = 0.866; p < 0.0001) for the model trees (Figure 4a). Thus, the model should work better for a community than for a single tree.

It is considered that the most visible changes in crown and trunk size concern data collected from a wide range of stand densities, which could be a problem when fitting equations [76]. It seems, therefore, that the inclusion of the density index as a predictive variable in this model was highly beneficial. The developed $H_{cal}^{*}$ model could work well in overpopulated and thinned communities. The analysis showed that “the square of the crown length is directly proportional to the geometric mean height and diameter of the tree at the ground level”. Such a relationship is a representation of biological processes occurring for many years in the phytocenose of pine, which can be expressed as: “the square of the length of the tree crown is equivalent to the leaf area index, and the geometric mean height and diameter express the effect of assimilation”.

The development of the proposed crown length model is based on the assumption that the crown is formed by whorls built of six branches. While measuring the length of the crowns in the sample trees, their length was reduced depending on the number of branches missing from the whorl (Figure 1). This is a new approach to the definition of crown length, as it is based on biological rather than technical considerations. The proposed model allows the crown length to be calculated for both individual trees and stands. DBH, height, and density are the main variables at the tree and stand level, respectively, and are used extensively around the world in forest inventories and forest management planning. This fact speaks for a wider use of the proposed model in forestry practice.

5. Conclusions

In this article, we proposed the allometric model of crown length determination for Pinus sylvestris L., which uses easy-to-measure biometric parameters, i.e., DBH, tree height, and density. It proved highly compatible with the actual crown lengths, as evidenced by the coefficient of determination r² = 0.95. The model is based on empirical data collected from 300 sample trees growing at different densities (from $z$ = 0.31 to $z$ = 1.86) and different ages (30 to 38 years). The development of an empirical crown length model is an important task of applied ecology. From an economic point of view, this model can be used to predict the mechanical stability of trees and their susceptibility to wind and snow damage. As it turned out, the equation benefited from including the density index, which mainly determines the amount of light reaching the lower branches of the crown and whether the needles remain on the tree or fall. By developing the allometric crown length model, a very useful indirect tool for estimating foliage size was provided, as it could determine the effectiveness of the physiological processes responsible, among others, for the growth of the tree. The precise determination of the crown length could, therefore, be one of the components used to estimate the mass of needles or leaf area index (LAI) and, consequently, the amount of transpiration or the amount of carbon dioxide bound, which is crucial in the context of climate change. Our further research work will focus on developing models to calculate the mass and leaf area index, as well as the amount of transpiration in pine monocultures, and the crown length model will be one of the most important components in this modelling.