Modelling the Tree Height, Crown Base Height, and Effective Crown Height of Pinus koraiensis Plantations Based on Knot Analysis

Abstract

Taking 1735 Pinus koraiensis knots in Mengjiagang Forest Farm plantations in Jiamusi City, Heilongjiang Province as the research object, a dynamic tree height, effective crown height, and crown base height growth model was developed using 349 screened knots. The Richards equation was selected as the basic model to develop a crown base height and effective crown height nonlinear mixed-effects model considering random tree-level effects. Model parameters were estimated with the non-liner mixed effect model (NLMIXED) Statistical Analysis System (SAS) module. The akaike information criterion (AIC), bayesian information criterion (BIC), −2 Log likelihood (−2LL), adjusted coefficient (R_a²), root mean square error (RMSE), and residual squared sum (RSS) values were used for the optimal model selection and performance evaluation. When tested with independent sample data, the mixed-effects model tree effects-considering outperformed the traditional model regarding their goodness of fit and validation; the two-parameter mixed-effects model outperformed the one-parameter model. Pinus koraiensis pruning times and intensities were calculated using the developed model. The difference between the effective crown and crown base heights was 1.01 m at the 15th year; thus, artificial pruning could occur. Initial pruning was performed with a 1.01 m intensity in the 15th year. Five pruning were required throughout the young forest period; the average pruning intensity was 1.46 m. The pruning interval did not differ extensively in the half-mature forest period, while the intensity decreased significantly. The final pruning intensity was only 0.34 m.

1. Introduction

The crown size of a tree reflects its photosynthetic ability and contains ecological legacy information related to the past history of the individual tree; the crown size can thus provide a reference for present and future tree growth [1,2,3,4]. The height to the crown base is defined as the height from the ground to the base of the first normal, live branch, and this metric is a useful indicator of tree vigour, wood quality, and wind firmness [5,6,7,8]. The tree crown can be classified into the “effective crown” and “non-effective crown” in the vertical direction based on whether the branches can contribute excess photosynthetic products for trunk growth [9,10]. Quantifying the height to the effective crown can provide reasonable pruning guidance and ultimately improve the timber quality of trees [11]. However, most of the existing studies on the height to the crown base and height to effective crown have been based on static measurements. Although the dynamic variations in the height to crown base and height to effective crown can be remeasured within equal intervals, these measurements are relatively difficult and time consuming to obtain, and the information provided by these measurements is still limited [12]. Trees prune naturally, and branches break at random locations, resulting in branches with varying sizes remaining on the stem. The stem continues to retain the remaining parts of broken branches and easily forms large knots to affect the wood material; thus, artificial pruning can be conducted before the onset of natural pruning to prevent these processes. The knots in a stem can yield abundant information related to crown structure dynamics throughout the lifetime of a tree, and the height to the crown base and height to the effective crown can also be accurately quantified using stem knots [13]. Therefore, in the present study, we mainly focused on modelling tree height, height to the crown base, and heights to effective crown using the knot-analysis approach.

Dynamic changes in the crown base height refer to the process in which the first living branch at the base of the canopy dies due to shading from the upper branch, and the height of the first living branch increases accordingly; this process is also known as crown recession [9]. Many studies have focused on the crown base height model and have achieved many findings. Rijal [14] used the index model and logistic model as the research basis, selected the tree height and diameter at breast height as the most important variables in the crown base height models, and analysed and compared the two resulting model forms. They believed that the logistic model had the best effect in predicting the crown base height. Based on the logistic model, Duan [15] fitted a mixed-effects crown base height model and found that the higher the growth advantage of a tree was, the higher the crown base height was. The dynamic changes in the height to the effective crown are a function of the photosynthetic contributions of branches to the crown. A effective crown is a crown that plays a major role in the growth of the tree, that is, the crown comprising all effective branches [16]. In recent studies, Geng [17] extracted the effective crown of an entire tree stand using airborne light detection and ranging (LiDAR) measurements. In addition, unmanned aerial vehicle (UAV)-derived LiDAR has also beenused to obtain crown base height measurements, such as in the study of Liu [18]. The part of a tree between the effective crown height and the crown base height is the non-effective crown; this tree section has less of an effect on tree growth. Therefore, when pruning timber trees, the goal is to eliminate the non-effective crown and leave the effective branches that affect the growth of the tree to ensure the growth vigour of the tree. However, most developed models have been based on static or dynamic measurements.

Compared to the use of static or dynamic growth methods for crown base height modelling, knot analysis technologies have been used to analyses crown dynamics [19,20,21]. Magrrie and Hann [22] used a stem analysis technique to calculate the heights of dead branches and the years of tree deaths, obtain historical crown base heights of the studied trees, and determine the dynamic changes in the corresponding crown bases within 5 years. Kershaw et al. [23] also studied the longevity and duration of radial growth in Douglas fir branches using knot analysis techniques. Jia and Chen [12] developed dynamic crown base height models for Larix olgensis using a nonlinear mixed effects model based on branch mortality analysis. Korean pine (Pinus koraiensis) plantations have been established in many areas in northeast China for the purposes of harvesting high-quality timber and producing seeds [24,25]. However, studies related to developing models of the overall heights of trees, crown base heights, and heights to effective crowns are still limited at present. The two main purposes of this study are: (1) Find the best models to estimate crown base height and height to the effective crown of Korean pine, by means of knot analysis technology. (2) Propose a reference for pruning strategies in forest management research, by providing a simulation of possible time and artificial pruning intensity based on the crown base height and height to the effective crown developed models.

2. Study Area

The study area is the Mengjiagang Forest Farm in Jiamusi City, Heilongjiang Province (130°32′42″~130°52′36″ E, 46°20′~46°30′50″ N) [26]. This region belongs to the continental monsoon climate zone of east Asia, with an average temperature of 2.7 °C. The extreme maximum temperature in the study region is 35.6 °C, the lowest temperature is −34.7 °C, the annual average precipitation is 550 mm, the annual sunshine hours are 1955 h, and the frost-free period lasts approximately 120 d. The forest site is located in the western part of Wandashan Mountain and is dominated by low hills with gentle slopes; most slopes in the study area are between 10° and 20°. The analysed forest site is within the Songhua River tributary system, and the largest river is the Liushu River, a third-level tributary of the Songhua River. The soil is dominated by dark brown soil, and typical dark brown soils are distributed most widely. The zonal vegetation belongs to the Xiaoxing’anling-Zhangguangcailing subregion, and the forest farms are dominated by artificial coniferous forests. The main species are Larix olgensis Henry, Pinus koraiensis Sieb. et Zucc, and Pinus sylvestris L. var mongolica. The area of planted forests is 9482 hm², and the stand stock is 1,005,400 m³.

3. Materials and Methods

3.1. Data Collection

The data used in the present study were collected from 12 planted Korean pine sample plots that were developed in different forest stands in the Mengjiagang Forest Farm in 2010; each plot had an area of 0.06 hm². All the trees in each plot were numbered, and the diameter at the breast height (DBH), tree height (HT), height to the crown base (HCB, in m), and crown width (CW, in m) of each tree were measured. Based on the investigations of the sample plots, the trees were divided into 5 classes (according to the order of diameter class or DBH, the trees are divided into 5 diameter classes with equivalent basal areas, and the quadratic mean DBH of each class is calculated which is 5 classes) using equivalent basal areas [27]. The average DBH and HT values of trees in each class were calculated separately, and 5 pines of corresponding sizes were selected from each plot. Thus, a total of 60 trees were selected with which to conduct stem and knot analyses. Below the crown base of each tree, near a knot, a 10~30-cm-long wooden section was intercepted and brought back to the laboratory, and the knots collected during each round were numbered. The height of each knot from the ground (calculated using the height of the knot in the tree segment and the height of the round branch) was measured, and the azimuth of each knot was measured according to the method by which the azimuths of the branches were measured (the direction of due north is 0 degrees and rotates clockwise). The position of each knot could be determined by analysing the scar left on the stem after the branch died. The two largest knots in each wood segment were selected, and a hand-held chainsaw was used to longitudinally dissect the stem, passing through the stem core, to obtain a longitudinal section of each knot. Each of these sections was scanned, and the number of annual rings in each knot was counted using an annual image analysis system (WinDENDROTMV6.5) [28]. Using the methods described by Fujimori [29] and Harri [30], a knot analysis was performed by determining the following values: (1) the age at which each branch grew outward from the medulla, (2) the age at which each branch completely stopped growing, (3) the age at which each branch had no growth vigour, and (4) the age at which each branch was completely occluded. Therefore, in this experiment, the branch diameter (BD), azimuth ($\varnothing$), age at birth (RB), age at which growth stopped (RBC), age of death (RBD), and age of full occupation (RBO) were obtained for a maximum of 2 knots per round of analysing 60 trees (Figure 1); through this data collection, a total of 1767 knot data points were obtained. Due to mistakes in the manual analysis process, some knots were broken; thus, a total of 1734 unbroken knots were obtained, and because these knots were associated with low growth vigour, the knots were not dead or not hidden. Therefore, death ages could be determined for only 1601 knots, and only 102 knots were completely occluded. The definitions of the relevant variables are listed in

Table 1. Descriptive information of the symbols used in this study.

Variable Symbol	Description
HT	Tree height (m)
HEC	Height of the effective tree crown (m)
HBC	Height of the tree crown base (m)
ECL	Length of the effective tree crown (m)
CL	Length of the living crown (m)
RCL	Ratio of the crown length
RB	Age at the time of the birth of the knot (a)
RBC	Age at the time the knot stopped growing (a)
RBD	Age at the time of death of the knot (a)
RBO	Age at the time when the knot was occluded (a)

. The main forest stand factors representing each standard plot, as well as the analytical tree and knot data statistics applied in the modelling and inspection processes, are listed in

Table 2. Statistics of stand variables, sampled trees, and knot distributions measured in Korean pine plantations.

Variable			Sample Size	Min	Max	Mean	Std (Standard Deviation)	Coefficient of Variation (CV%)
Fitting dataset	Plot variable	Age (a)	10	37	47	40.9	3.60	8.81
		Density (trees·hm−2)	10	650	1533	991.1	296.45	29.91
		Elevation (m)	10	201.5	262.8	242.5	18.73	7.72
	Tree variable	DBH (cm)	50	12.5	27.3	20.5	3.52	17.16
		HT (m)	50	9.5	16.1	12.5	1.30	10.39
		CL (m)	50	1.9	9.9	6.3	1.65	26.38
		CW (m)	50	1.4	3.7	2.3	0.63	26.71
		H/D	50	0.4	1.0	0.6	11.36	18.20
		RB	1445	1	30.0	11.6	5.79	50.10
		RBC	1445	7	46.0	21.8	6.34	29.03
		RBD	1320	9	49.0	27.6	6.93	25.16
		RBO	95	16	46.0	31.8	6.19	19.49
Validation dataset	Plot variable	Age (a)	2	32	37	34.5	3.54	10.25
		Density (trees·hm−2)	2	1467	1650	1558.5	129.40	8.30
		Elevation (m)	2	194.6	222.6	208.6	19.80	9.49
	Tree variable	DBH (cm)	2	12.3	19.5	16.1	2.50	15.58
		HT (m)	2	9.6	11.3	10.3	0.58	5.64
		CL (m)	2	3.5	6.6	5.3	0.88	16.70
		CW (m)	2	1.3	2.4	1.7	0.35	20.64
		H/D	2	0.5	0.8	0.7	0.10	14.85
		RB	290	1	20	9.8	4.92	50.06
		RBC	290	8	33	18.3	4.88	26.73
		RBD	282	11	36	24.5	6.13	25.08
		RBO	7	24	36	30.6	4.39	14.36

H/D: Tree height/diameter at breast height.

.

3.2. Model Development

3.2.1. Data Screening

Based on the knot data analysis research of Sprugel, Fujimori, and Li [22,31,32], it can be concluded that the photosynthetic production of a branch is weakened when the branch stops growing, and the branch can thus no longer affect the growth of the stem. Therefore, all the branches above the height of the position where the branches have stopped growing constitute the effective crown. The heights at which branches are located when their growth stops can be applied as the critical height of the effective crown. Similarly, the HCB corresponds to the height of the critical crown at the time a branch died. The branches located between the effective crown height and the HCB cannot contribute to the growth of the stem but also cannot guarantee the sufficient supply of nutrients needed for their own growth and development. In the utilized knot-selection method, when the RBC and RBD values of a certain section are larger than the RBC and RBD values of another knot at the corresponding height, the heights corresponding to the RBC and RBD ages of the branches forming the knot are considered to be the effective crown height and crown base height, respectively. Through the above method, 349 knot data points were obtained to establish a dynamic effective crown height and crown base height growth model. The trees measured in 12 plots were randomly divided into model-fitting data and independent validation data. The first 10 plots were used for the model-fitting process, and the remaining 2 plots were used to validate the model.

3.2.2. Basic Model Selection

The main purpose of this paper was to develop a mixed-effects model for tree heights, effective crown heights, and crown base heights and to calculate the required pruning time and intensity based on the developed model. In the past, many types of tree height, effective crown height, and crown base height models have been used, such as the logistic equation, Mitscherlich equation, Gompertz equation, Richards equation, and Korf equation. Many studies have shown that the Richards equation is highly adaptable and accurate, and the parameters used in this equation are biologically significant and widely used in growth estimations [26]. The Richards equation model form is as follows:

$$H=a_1\times {\left(1-exp\left(-a_2\times t\right)\right)}^{a_3}+\epsilon$$

(1)

where H is the total tree height, height to the crown base, or height to the effective crown, $a_1$ is the maximum tree height, crown base height, or effective crown height parameter, $a_2$ is the growth rate parameter, $a_3$ is the shape parameter, t is the corresponding age, and $\epsilon$ is the random error.

The single-level nonlinear mixed model with all the parameters in model (1) set as random effects is shown in Equation (2):

$$H_{ij}=\left(a_1+b_{i1}\right)\times {\left(1-exp\left(-\left(a_2+b_{i2}\right)\times t_{ij}\right)\right)}^{\left(a_3+b_{i3}\right)}+{\epsilon }_{ij}$$

(2)

where $H_{ij}$ is the tree height, effective crown height, or the crown base height in the jth year of the lifetime of the ith tree; $t_{ij}$ is the age at which the knot corresponding to $H_{ij}$ stops growing or dies; $a_1$, $a_2$, and $a_3$ are fixed-effect parameters; $b_{i1}$, $b_{i2}$, and $b_{i3}$ are random parameters in the model; and ${\epsilon }_{ij}$ is the error term of the model.

3.2.3. Mixed-Effects Model Development

Mixed-effects models contain fixed and random-effects parameters, where fixed-effects parameters are common to all trees in the sample and random-effects parameters are specific to each tree [33]. In this study, the basic and mixed models were fitted with combinations of 7 different random parameters based on the random effects of trees. The mixed-effects model was developed using the following four steps.

• To construct a mixed-effects model based on a basic model, we first needed to determine the categorical variables. All single-level models consider the corresponding levels to be categorical random-effects variable.
• The random parameters in the Richards model were determined. The NLMIXED module of the Statistical Analysis System (SAS 9.2) [34] software was used to fit the single-level effective crown height and crown base height hybrid model, and random combinations of different random parameters were considered to determine the best nonlinear hybrid model.
• The variance-covariance random-effect structure generally adopts the generalized positive definite matrix D; this matrix mainly reflects the differences among the tree samples and can be expressed as follows:

  $$D=\begin{array}{ccc}{\sigma }_{b_{i1}}^2& {\sigma }_{b_{i1}{b}_{i2}}& {\sigma }_{b_{i1}{b}_{i3}}\\ {\sigma }_{b_{i2}{b}_{i1}}& {\sigma }_{b_{i2}}^2& {\sigma }_{b_{i2}{b}_{i3}}\\ {\sigma }_{b_{i3}{b}_{i1}}& {\sigma }_{b_{i3}{b}_{i2}}& {\sigma }_{b_{i3}}^2\end{array}$$

  (3)

  where ${\sigma }_{b_{i1}}^2$ is the variance in random parameter $b_{i1}$, ${\sigma }_{b_{i2}}^2$ is the variance in random parameter $b_{i2}$, ${\sigma }_{b_{i3}}^2$ is the variance in random parameter $b_{i3}$, ${\sigma }_{b_{i1}{b}_{i3}}={\sigma }_{b_{i3}{b}_{i1}}$ is the covariance between random parameters $b_{i1}$ and $b_{i3}$, ${\sigma }_{b_{i1}{b}_{i2}}={\sigma }_{b_{i2}{b}_{i1}}$ is the covariance between random parameters $b_{i1}$ and $b_{i2}$, and ${\sigma }_{b_{i2}{b}_{i3}}={\sigma }_{b_{i3}{b}_{i2}}$ is the covariance between the random parameters $b_{i2}$ and $b_{i3}$.
• The intragroup variance-covariance structure, A, is determined. To determine the within-group variance structure, the heteroscedasticity and autocorrelation problems must be solved. The variance structure of the residuals must be considered. Therefore, the intragroup variance-covariance can be expressed as follows:

  $$R_i={\sigma }^2{I}_{ni}$$

  (4)

  where ${\sigma }^2$ refers to the residual variance value of the observed object and $I_{ni}$ is the $n_i\times n_i$-dimensional unit matrix describing the change.

3.3. Model Evaluation

The NLMIXED module was used in SAS software to estimate the mixed-effects model parameters. The Akaike information criterion (AIC), Bayesian information criterion (BIC), and deviance (−2LL) were used to select the optimal model. To avoid the excessive parameterization problem, models with different numbers of parameters were also compared using the likelihood ratio test (LRT). If p < 0.005, the difference was considered significant. The results of three evaluation indexes, the adjusted coefficient (Ra²), root mean square error (RMSE), and residual squared sum (RSS), were used to evaluate the modelling effects. The larger the Ra² value was and the smaller the RMSE value was, the better the model performance was. The independent validation sample data were used to test the established model and the mean error (ME), mean absolute error (MAE), relative mean error (M%E), relative absolute mean error (MA% E), estimation accuracy (P) were used as statistical indicators to evaluate the model performance.

3.4. Time and Intensity of Artificial Pruning

Time and intensity of artificial pruning is a simulation based on the previous developed models [35]. Using the tree height, crown base height, and effective crown height model obtained as a function of time in this study, as established with the collected knot data, three equations were plotted. The effective crown height and crown base height values were calculated, and the time at which the effective crown height and crown base height value were large and negative was considered as the initial pruning time (t₀); then, t₀ is introduced into the crown base height model to calculate the corresponding height (h₀), which reflects the height at which the pruning starts. Since the effective crown is the part of the crown that plays a major role in tree growth, t₀ can be introduced to the established effective crown height model to calculate the height h₁ at the corresponding time. The time of the first pruning is t₀, and the pruning intensity is h₁–h₀. The starting height of the second pruning is at the position h₁ at the end of the last pruning. The pruning height h₁ is introduced into the sub-height model to calculate the pruning time t₁; then, t₁ is introduced into the effective crown height model. At this time, the effective crown height is h₂, and the pruning intensity is h₂–h₁. This process is repeated until the Korean pines reach maturity and the entire rotation period of the Korean pine plantations ends. The equations used to calculate pruning time and intensity were showed as follows.

$$\Delta h_i=h_i-h_{i-1}$$

(5)

$$\Delta t_i=t_i-t_{i-1}(i>1)$$

(6)

where ∆h_i is pruning intensity, h_i is effective crown height, ∆t_i is time interval between pruning, t_i is the ith pruning time, i is the number of pruning (i = 1, 2, …, n).

4. Results and Analysis

4.1. Tree Height Model Selection

Based on a previous study, the Richards equation was used herein as the basic model to develop a mixed-effect model for the total tree height.

The parameter estimates and goodness-of-fit statistics are listed in

Table 3. Fitting results of the mixed-effects model based on individual tree height effects.

Model Form	Random Effects Considered	Number of Simulations	AIC	BIC	−2LL	LL	LRT	p
0	None	4	5142.7	5164.9	5134.7	−2567.35	-	-
1	b1	5	5034.4	5032.4	5024.4	−2512.2	110.3	<0.0001
2	b2	5	5039.3	5037.2	5029.3	−2514.65	-	-
3	b3	5	5060.3	5058.4	5050.3	−2525.15	-	-
4	b1, b2	7	5029.4	5026.6	5015.4	−2507.7	9.0	0.0111
5	b1, b3	7	5031.6	5028.8	5017.6	−2508.8	-	-
6	b2, b3	7	5103.1	5100.3	5089.1	−2544.55	-	-
7	b1, b2, b3	10	-	-	-	-	-	-

p value showed significant difference between models. When p < 0.05, it is considered that the difference between models is significant.

. When the model does not contain random parameters, the AIC value was 5142.7, the BIC value was 5164.9, and the −2LL value was 5134.7. When considering all seven different random parameter combinations, the model with three random parameters did not converge. The AIC, BIC, and −2LL values of the six models that converged were smaller than those of the model without random parameters. Among the models in which one random parameter was added, as shown in the table below, the effect was optimized when random parameter b1 was added. When two random parameters were added, model with the b1 and b2 parameter combination has the smallest AIC, BIC, and −2LL values. The LRT results indicated that the basic model differed significantly from the hybrid model when random parameter b1 was added (LRT = 110.3, p < 0.0001). The model with one random parameter (b1) and the model with two random parameters (b1 and b2) were not significantly different (LRT = 9, p = 0.0111). Based on the above analyses, the tree height hybrid model that considered random tree effects performed better than the basic model, and the tree height hybrid model containing random parameter b1 was the optimal model.

4.2. Effective Crown Height Model Selection

The results of the effective crown height model fit are shown in

Table 4. Fitting results of the mixed-effects model based on individual effective tree crown height effects.

Model Form	Random Effects Considered	Number of Simulations	AIC	BIC	−2LL	LL	LRT	p
8	None	4	899.6	914.3	891.6	−445.8	-	-
9	b1	5	686.0	695.6	676.0	−338.0
10	b2	5	679.7	689.3	669.7	−334.9	221.9	<0.005
11	b3	5	683.8	693.3	673.8	−336.9
12	b1, b2	7	658.7	672.1	644.7	−322.4
13	b1, b3	7	650.8	664.2	636.8	−318.4	32.9	<0.005
14	b2, b3	7	848.7	862.1	834.7	−417.4
15	b1, b2, b3	-	-	-	-	-	-	-

p value showed significant difference between models. When p < 0.05, it is considered that the difference between models is significant.

. When no random parameters were considered in the model, the AIC value was 899.6, the BIC value was 914.3, and the −2LL value was 891.6. The model did not converge when considering combinations of three different random parameters. When one random parameter was added, the AIC, BIC, and −2LL values were all reduced. It can be seen from the table that the effect was best when the random parameter b2 was added. When two random parameters were added in combinations of b1/b2 and b1/b3, the AIC, BIC, and −2LL values continued to decrease, and the values obtained with the b1/b3 combination were the smallest. When the b2/b3 variable combination was considered, the AIC, BIC, and −2LL values were smaller only than those of models with no random parameters. The LRT results showed that the basic model and the model considering random parameter b2 were significantly different (LRT = 221.9, p < 0.005), and the model considering the random parameter b2 was also significantly different from the model considering two random parameters (b1 and b3) (LRT = 32.9, p < 0.005). Based on the above analyses, the effective crown height mixed effects model that considered tree effects was better than the basic model, and the mixed effects model that considered two random parameters performed better than the mixed effects model that considered only one random parameter.

4.3. Crown Base Height Model Selection

The crown base height model simulation method was the same as that applied to the effective crown height model. When the model did not consider random parameters, the AIC value was 911.2, the BIC value was 925.8, and the −2LL value was 903.2. The model did not converge when considering combinations of three different random parameters among seven total parameters. When one random parameter was added, the AIC, BIC, and −2LL values were all reduced.

Table 5. Fitting results of the mixed-effects model based on individual tree crown base height effects.

Model Form	Random Effects Considered	Number of Simulations	AIC	BIC	−2LL	LL	LRT	p
16	None	4	911.2	925.8	903.2	−451.6	-
17	b1	5	713.1	722.7	703.1	−351.55	200.1	<0.005
18	b2	5	718.8	728.4	708.8	−354.4
19	b3	5	743.9	753.4	733.9	−366.95
20	b1, b2	7	735.8	749.2	721.8	−360.9
21	b1, b3	7	703.8	717.2	689.8	−344.9	13.3	<0.005
22	b2, b3	7	848.1	861.5	834.1	−417.05
23	b1, b2, b3	-	-	-	-	-	-	-

p value showed significant difference between models. When p < 0.05, it is considered that the difference between models is significant.

shows that the modelling effect was optimized when the random parameter b1 was added. When two random parameters were added (b1/b2 and b1/b3), the AIC, BIC, and −2LL values continued to decrease; the values were smallest when the b1/b3 combination was considered. When the considered variable combination was b2/b3, the AIC, BIC, and −2LL values were smaller than those of the model that considered no random parameters. The LRT results showed that the basic model was significantly different from the model that considered the single random parameter b1 (LRT = 200.1, p < 0.005). The model that considered b1 was also significantly different from the model that considered two random parameters (b1 and b3) (LRT = 13.3, p < 0.005). Based on the above analyses, the crown base height hybrid model in which tree effects were considered performed better than the basic model, and the hybrid model that considered two random parameters performed better than the hybrid model that considered only one random parameter.

4.4. Model Evaluation

Table 6. Fixed parameters and variance component estimates of models with different numbers of considered parameters.

Model	Response Variable	a1	a2	a3	Covariance-Structure D	Ri	Ra2	RMSE	RSS
Model 0	HT	15.4674	0.0622	2.5315	-	-	0.9496	0.9283	0.6573
Model 1	HT	15.5153	0.0619	2.5239	$\left[0.2424\right]$	0.8047	0.9529	0.8971	0.6351
Model 4	HT	15.5400	0.0618	2.5213	$\left[\begin{array}{cc}0.4845& -0.00125\\ -0.00125& 0.000006313\end{array}\right]$	0.7979	0.9534	0.8930	0.6311
Model 8	HEC	10.5977	0.0716	4.2887	-	-	0.7132	1.1335	367.4293
Model 10	HEC	12.1014	0.0687	4.5563	$\left[0.000089\right]$	0.3754	0.9286	0.5656	91.4864
Model 13	HEC	11.4905	0.0726	4.7620	$\left[\begin{array}{cc}8.2599& 2.0943\\ 2.0943& 1.5803\end{array}\right]$	0.2743	0.9390	0.5225	77.2625
Model 16	HBC	10.9893	0.0593	5.0751	-	-	0.7015	1.1562	382.3540
Model 17	HBC	9.7940	0.0717	6.6491	$\left[4.9432\right]$	0.4407	0.9156	0.6149	108.1239
Model 21	HBC	9.2716	0.0792	7.8990	$\left[\begin{array}{cc}3.7297& 0.7670\\ 0.7670& 2.9486\end{array}\right]$	0.3683	0.9349	0.5402	82.5704

provides the fixed parameter estimates obtained for the effective crown height model, crown base height model, and optimal mixed model under different parameter combinations, as well as the variance-covariance compositions of the random effects, the intragroup variance-covariance structures, and the adjusted values of three evaluation indicators: Ra², RMSE, and RSS. After the tree effects were introduced, the determination coefficient of the mixed-effects model was larger than that of the basic model, and the RMSE and RSS values of the hybrid model were smaller than those of the basic model. These evaluation indicators show that the introduction of random parameters improved the fitting accuracy of the models. In both the effective crown height model and the crown base height model, the introduction of two random parameters can improve the fitting accuracy of the models more than the introduction of one random parameter. The residual maps of the effective crown height and the underlying high values obtained from the basic model (model 8 and model 16) and optimal hybrid model (model 13 and model 21) are plotted separately (Figure 2). Figure 2 also shows that the accuracy of the mixed model that considered random tree effects was significantly higher than that of the basic model.

The remaining sample data were verified by calculating the ME, MAE, average relative deviation (M%E), average relative deviation absolute value (MA% E), and estimation accuracy (p) values. The results are shown in

Table 7. Results of the model validations.

Model	Response Variable	ME	MAE%	p
Model 1	HT	−0.3301	0.1773	0.9893
Model 13	HEC	0.4196	0.2936	0.9227
Model 21	HBC	0.1363	0.2926	0.9466

.

4.5. Estimations of the Time and Intensity of Artificial Pruning

In this study, tree height, crown base height, and effective crown height prediction models were developed through stem and knot analyses and used to draw the graphs shown in Figure 2. Based on these fitted model, the simulated pruning process was predicted. The simulated process was showed as follows. The tree height, effective crown height, crown base height, crown length, effective crown length, and effective crown height and crown height difference values were calculated for the first 20 years of tree growth (

Table 8. Statistical values predicted for Korean pine plantations.

AGE (y)	HT (m)	HEC (m)	HBC (m)	ECL (m)	CL (m)	Difference (m)
1	0.02	0.00	0.00	0.02	0.02	0.00
2	0.08	0.00	0.00	0.07	0.08	0.00
3	0.19	0.01	0.00	0.18	0.19	0.01
4	0.36	0.03	0.00	0.33	0.35	0.02
5	0.58	0.06	0.01	0.51	0.57	0.05
6	0.84	0.12	0.02	0.72	0.81	0.09
7	1.14	0.20	0.05	0.95	1.09	0.15
8	1.48	0.30	0.08	1.17	1.40	0.22
9	1.84	0.43	0.12	1.40	1.72	0.31
10	2.23	0.60	0.19	1.63	2.04	0.41
11	2.63	0.78	0.26	1.85	2.37	0.52
12	3.05	1.00	0.36	2.05	2.69	0.64
13	3.47	1.23	0.47	2.24	3.00	0.76
14	3.91	1.49	0.60	2.42	3.31	0.89
15	4.34	1.76	0.75	2.58	3.59	1.01
16	4.78	2.05	0.92	2.73	3.86	1.14
17	5.21	2.35	1.10	2.86	4.11	1.25
18	5.64	2.66	1.29	2.98	4.34	1.36
19	6.06	2.97	1.51	3.09	4.55	1.46
20	6.47	3.29	1.73	3.19	4.74	1.56

). As shown in Figure 3 and Table 8, the effective crown height and crown base height differ by more than 1 metre at the 15th year. Therefore, the first simulated pruning time begun at the 15th year. The simulated intensities and times of the next 10 pruning were estimated using the initial pruning time, as shown in

Table 9. Statistical values obtained for the pruning time and intensity for Korean pine plantations.

Frequency	AGE (y)	HT (m)	HBC (m)	HEC (m)	Intensity (m)	Interval (y)	Crown Length Ratio before Pruning (%)	Crown Length Ratio after Pruning (%)
1	15	4.34	0.75	1.76	1.01	0	0.83	0.59
2	20	6.47	1.76	3.29	1.52	5	0.73	0.49
3	26	8.38	3.29	5.13	1.85	6	0.61	0.39
4	32	10.54	5.13	6.71	1.57	6	0.51	0.36
5	39	12.13	6.71	8.08	1.37	7	0.45	0.33
6	46	13.24	8.08	9.01	0.93	7	0.39	0.32
7	53	14.01	9.01	9.61	0.60	7	0.36	0.31
8	59	14.46	9.61	9.95	0.34	6	0.34	0.31
9	63	14.69	9.95	10.11	0.16	4	0.32	0.31
10	66	14.83	10.11	10.20	0.09	3	0.32	0.31

. According to the age group classification, Korean pine plantations were considered young forests when the trees were less than or equal to 40 years old, middle-aged forests when the trees were 41 to 60 years old, near-mature forests when the trees were 61 to 80 years old, and mature forests when the trees were more than 80 years old. In the young forest period, growth was faster, the crown lengths were relatively large, the simulated average pruning interval was 6 years, and the simulated average pruning intensity was 1.46 metres. When the plantations entered the middle-aged forest stage, the crown lengths decreased to below 0.4 metres, and the pruning interval did not differ extensively from that of the young forests; however, the pruning intensity decreased significantly, and the last pruning intensity was only 0.34 metres. Considering the actual production and management situations in Korean pine plantations, it is basically unnecessary to prune Korean pines after plantations enter the near-mature forest stage. The crown length ratios of Korean pines stabilize at approximately 0.3 metres on the 9th and 10th pruning, as shown in Table 9. Therefore, based on the developed model and the simulated pruning process for the Korean pine plantations in the Mengjiagang area, only young and middle-aged forests need to be pruned. The time, the simulated interval, and intensity of the pruning can be found in Table 9.

5. Discussion and Conclusions

The crown base height of a tree determines the crown ratio of the tree and may further affect the crown structure, stem growth, and even biomass estimation [1,29]. Effective crown height data can provide scientific references for artificial pruning to achieve high-quality wood [10,12]. In forestry practice, the quality of an individual tree mainly depends on the number, location, and size of its branches [30]. Knot-related defects have always been the major factor leading to timber quality degradation, and avoiding this kind of defect is a hot topic in forest management research [31]. Therefore, determining the crown base height and effective crown height is essentially important. Knot characteristics can reflect the dynamic recession of the crown and, eventually, the production of clean wood [32]. The effective crown height is closely related to productivity estimations of individual trees and forests [36]. Thus, the effective crown height, along with the tree height and crown base height, are useful characteristics in individual tree growth and yield modelling systems.

Pruning is an important management strategy for producing high-quality, large-diameter timber in silviculture by regulating the distribution of photosynthates in individual trees [37]. In the present study, we developed a nonlinear mixed-effects crown base height and effective crown height model to determine the crown base and effective crown heights. The Pinus koraiensis trees are short, so deviations may arise when using this model to predict Korean pine plantations in other forest farms. When using the Richards model to establish a mixed-effects model, only the tree effects were added; plot effects were not taken into account. When the lower parts of the modelled tree canopies began to exhibit dead branches, pruning was appropriate. The height between the crown base and the splitting point between the effective crown and non-effective crown was calculated to determine the time of initial pruning. The initial pruning time was found to be 15 years, and this finding was consistent with the results obtained by Wang et al. [31], who reported that a 15-year-old Korean pine plantation entering a period of vigorous growth. The calculated results show that pruning activities are mainly concentrated in young and middle-aged forests, and young forests grow vigorously.

An important risk associated with green pruning involves branch wounds inflicted by wood-decaying fungi [38]. Therefore, pruning may cause adverse results that counteract its original purpose of improving timber quality; these adverse results can involve the formation of decay and discolouration [39]. Many other factors also affect the discolouration patterns of trees, including the type of pruning tool used, the branch status, and the duration of branch occlusion [40,41]. The time until complete branch occlusion is another important factor affecting the health and economic value of tree wood. Therefore, we also focused on the pruning time and the average intervals between prunings in our study. We concluded that pruning activities were mainly concentrated in young and middle-aged forests; pruning was not required after the plantations entered the near-mature forest stage. The intensity of each pruning was large, the average pruning interval was 6 years, and the pruning intensity was 1.46 metres; after entering the middle-aged forest stage, the pruning interval was unchanged, while the pruning intensity decreased. The pruning intensity of middle-aged forests was 0.34 metres. Li [42] used a similar method to study the effective crown height and pruning techniques of larch plantations, and the results were similar to those obtained in this paper. Young pine saplings are suitable for grown in areas with insufficient light conditions, as shown in the tree height, effective crown height, and crown base height curves; however, this is not considered in the pine forests of the Mengjiagang Forest Farm, where forests are directly planted in open space during afforestation.

Competition is another important variable that affects crown recession and dynamics. The strength and interval of pruning activities also reflect the competition status of tree stands to a certain extent. In young and middle-aged forests, individual trees grow rapidly. As the number of individual trees increases, their crowns overlap with each other. The more intense this competition is, the more branches die in the lower parts of the crowns, and the greater the pruning intensity and shorter the pruning interval must be [43]. The effects of neighbouring trees are of particular importance for modifying the distribution of branches within the crown and therefore influence the knot distribution with regard to the timber quality [44]. In the future, we will further study the effects of forest competition on the effective crown height and other crown features. As traditional knot analysis approach is quite time consuming, we will further explore other advanced data-collection approaches to enlarge our dataset.