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Article

Development of Full Growth Cycle Crown Width Models for Chinese Fir (Cunninghamia lanceolata) in Southern China

by
Zheyuan Wu
1,
Dongbo Xie
1,
Ziyang Liu
1,
Linyan Feng
1,
Qiaolin Ye
2,
Jinsheng Ye
3,
Qiulai Wang
3,
Xingyong Liao
4,
Yongjun Wang
4,
Ram P. Sharma
5 and
Liyong Fu
1,*
1
Research Institute of Forest Resource Information Techniques, Chinese Academy of Forestry, Beijing 100091, China
2
State Key Laboratory of Tree Genetics and Breeding, Co-Innovation Center for Sustainable Forestry in Southern China, College of Information Science and Technology & Artificial Intelligence, Nanjing Forestry University, Nanjing 210037, China
3
Guangdong Forestry Survey and Planning Institute, 338 Guangshanyilu, Guangzhou 510520, China
4
Chengdu Academy of Agriculture and Forestry Sciences, Chengdu 610000, China
5
Institute of Forestry, Tribhuvan University, Kathmandu 44600, Nepal
*
Author to whom correspondence should be addressed.
Forests 2025, 16(2), 353; https://doi.org/10.3390/f16020353
Submission received: 31 December 2024 / Revised: 26 January 2025 / Accepted: 14 February 2025 / Published: 16 February 2025
(This article belongs to the Special Issue Forest Biometrics, Inventory, and Modelling of Growth and Yield)
Browse Figures
Figure 1
<p>Sample point distribution map.</p> "> Figure 2
<p>Methodology framework.</p> "> Figure 3
<p>Pearson’s correlation coefficients and collinearity tests among multiple variables and crown width: (<b>a</b>) correlation coefficients of CW with other variables; (<b>b</b>) collinearity of CW with each variable, where the y−axis shows the VIF values. CW, crown width; A, stand age (year); SD, stand density; CD, canopy closure; DBH, diameter at breast height; H, tree height; HCB, height to crown base; DH, dominant tree height.</p> "> Figure 4
<p>Comparison of model performance for predicting Chinese fir crown width (<span class="html-italic">RMSE</span>, Root Mean Square Error; <span class="html-italic">TRE</span>, Total Relative Error; blue points, training dataset; orange points, validation dataset).</p> "> Figure 5
<p>Distribution of residuals for three models predicting the CW of Chinese fir trees.</p> ">
Versions Notes

Abstract

:
This study focused on 16,101 Cunninghamia lanceolata trees across 133 plots in seven cities of Guangdong Province, China, to develop a comprehensive full growth cycle crown width (CW) model. We systematically analyzed the dynamic characteristics of CW and its multi-scale influencing mechanisms. A binary basic model, with the diameter at breast height (DBH) and height (H) as core predictor variables, effectively reflected tree growth patterns. The inclusion of age groups as dummy variables allowed the model to capture the dynamic changes in CW across different growth stages. Furthermore, the incorporation of a nested two-level nonlinear mixed-effects (NLME) model, accounting for random effects from the forest block- and sample plot-level effects, significantly improved the precision and applicability of the final model (R2 = 0.731, RMSE = 0.491). This model quantified both macro- and micro-level effects of region and plot on CW. Our findings showed that the two-level NLME model, incorporating tree age groups, optimally accounted for environmental heterogeneity and tree growth cycles, resulting in the best-fitting statistics. The proposed full growth cycle CW model effectively enhanced the model’s efficiency and predictive accuracy for Cunninghamia lanceolata, providing scientific support for the sustainable management and dynamic monitoring of plantation forests.

1. Introduction

Crown width, as a key parameter for measuring tree canopy size, plays a crucial role in forest management and individual-tree studies [1]. It not only directly reflects tree growth status but also serves as an important input variable in growth and yield models, such as aboveground biomass and basal area increment models [2,3]. In addition, crown width is used to assess tree health and vitality [4]; by monitoring changes in crown size and morphological features, it provides valuable insights into the physiological condition of trees [5]. The size and structure of the canopy are closely related to the photosynthetic efficiency of the forest [6], influencing the carbon sequestration capacity and energy flow of the entire forest community [6], thus affecting its climate regulation functions [7]. Variations in crown structure contribute to the maintenance of forest species diversity by providing diverse habitats for various organisms, thereby enhancing the stability and resilience of the ecosystem [8,9]. Accurate measurement and prediction of crown width are essential for developing scientific forest management plans in disciplines such as forest ecology, silviculture, and forest management. These plans include management practices such as appropriate thinning, breeding of superior tree species, and optimization of forest structure [8,10,11,12,13]. Furthermore, detailed information on tree crowns helps to understand the characteristics of ecosystem management, including forest productivity, biodiversity, and wildlife habitats [14]. Through the monitoring and analysis of crown width over the entire growth cycle, it is possible to effectively assess the health and service functions of forest ecosystems, thereby supporting the development of sustainable forest management and ecological conservation strategies [15]. This is critical for promoting the sustainable use of forest resources and optimizing ecological functions [11,16]. Although crown width is an important parameter for reflecting tree growth status [17], measuring it individually in all plots is time-consuming, labor-intensive, and costly [18,19]. Developing a high-precision crown width prediction model across the entire growth cycle has become a core focus of current research.
Traditional CW prediction models primarily rely on statistical methods to establish quantitative relationships between CW and various factors such as DBH, H, and stand density (SD) [20,21]. With the advancement of research, scholars have progressively incorporated multiple regression analyses and nonlinear models to better capture the complex interactions between CW and multiple influencing variables [22,23,24,25]. These models have evolved from ordinary least squares (OLS) regression and linear models to more sophisticated nonlinear models [26,27,28], and eventually to mixed-effects modeling that accounts for both fixed and random effects [22,23,24,25]. This progression has significantly improved model accuracy and applicability. However, despite the success of these models in short-term or specific growth stage predictions, they often overlook the variations in CW across different age stages. This limitation may affect their predictive accuracy and applicability when modeling the entire growth cycle. CW is not only a key visual parameter reflecting tree growth and productivity but also serves as an indicator of stand age and tree competition [29]. CW growth exhibits significant variations across different age stages [30]. Scholars have shown that before reaching near-maturity, tree CW generally increases with age, with the most pronounced growth occurring in young and middle-aged stands [31]. Several recent studies have accounted for these developmental differences by analyzing the effects of DBH and H on CW across different age stages, aiming to construct more accurate and rapid CW prediction models for Chinese fir trees [32,33]. Additionally, crown biomass allocation varies across different growth stages, further influencing CW estimation. For example, Mikšys et al. evaluated changes in aboveground biomass in pine trees across various stand ages and found that total crown biomass at the stand level remains relatively stable over time, whereas total aboveground biomass increases with stem biomass growth [34]. Similarly, Menéndez-Miguélez et al. investigated the variation in crown biomass ratios with tree size and identified three distinct patterns of crown-to-stem biomass allocation across different age stages [35]. These findings highlight the significant variations in CW across different stand ages, underscoring the necessity of incorporating age-related CW changes into full growth cycle prediction models for Cunninghamia lanceolata. By introducing age groups as dummy variables, it is possible to systematically analyze the influence of different age structures on CW dynamics. This approach enhances the reliability of full growth cycle predictions and ensures that the model effectively captures the growth characteristics of vegetation across different age groups [20,36].
Moreover, the differences in growth observed across various blocks and sample plots reflect the impact of diverse environmental factors, such as soil type, humidity, and climate conditions, which may significantly influence crown development [19,37]. The CW data required for the model are often acquired from different forest stands with varying growth conditions [38], leading to a hierarchical data structure (a sample plot nested in blocks), which may result in correlated observations. Previous studies have indicated that using OLS regression to estimate CW models with nested data structures will lead to biased results [39]. In contrast, mixed-effects modeling can account for much of the heterogeneity and randomness caused by both known and unknown factors [28], effectively separating the variation at the block and sample plot levels [38], thereby enhancing the model’s predictive accuracy. Researchers such as Chen [10] and Zhou [38] have successfully applied mixed-effects models to construct CW prediction models for different tree species, showing that mixed-effects models provide an effective approach in terms of prediction cost, model efficiency, and accuracy. However, current research primarily focuses on predicting crown width during a single growth stage or for short-term periods [40], neglecting the integration of multilevel environmental factors and age group changes over the full growth cycle. In this present study, we not only addressed the variable-independence issues arising from the nested data structure at the regional scale but also accounted for tree age by dividing the trees into five age groups and incorporating these groups into the CW model as dummy variables.
Cunninghamia lanceolata is not only a key fast-growing timber species in southern China but also accounts for 17.33% of the nation’s total plantation area—ranking first among plantation timber species—according to the Ninth National Forest Resources Inventory. Given its crucial role in maintaining ecological balance, enhancing forest carbon sinks, and increasing timber supply, the relationships between Chinese fir tree CW and its driving factors should be investigated. Addressing existing research problems in Chinese fir tree CW modeling, such as the lack of dynamic age group consideration, a hierarchical data structure, and noticeable autocorrelation and cross-correlation among observations, this study focused on pure Chinese fir plantations in southern China. Specifically, we (1) identified the most suitable function and introduced five age groups as dummy variables to develop a full growth cycle CW model and (2) constructed a two-level NLME CW model, considering the forest block- and sample plot-level effects as random effects. Drawing on a broad dataset of more than 16,000 samples, our NLME CW model effectively increases predictive accuracy, offering valuable insights and references for the sustainable management of Chinese fir plantations.

2. Materials and Methods

2.1. Study Area

Data were collected from 133 sample plots of pure Chinese fir plantation stands distributed in Lechang City, Yingde City, Heping County, Lianshan County, Longshan County, Yunan County, and Shixing County in Guangdong Province, China. Guangdong is situated in the most southern part of mainland China, spanning geographical coordinates from 109°45′ to 117°20′ E and 20°09′ to 25°31′ N. As of 2023, the province boasts a forested area of approximately 10.85 million hectares—57.1% of its total land area—with forest coverage extending over about 9.6 million hectares (53.9% coverage rate). This extensive forest area makes Guangdong one of the key ecological protection zones in southern China. The province’s terrain consists primarily of mountains, hills, plains, and water bodies, generally sloping downward from north to south. Guangdong experiences a subtropical monsoon climate characterized by distinct seasonal variations: warm and humid in spring, hot and rainy in summer, mild with less rainfall in autumn, and cool and dry in winter. Major vegetation types include southern subtropical evergreen broad-leaved forests and mid-subtropical evergreen broad-leaved forests, while the main soil types comprise lateritic soils, metamorphic soils, immature soils, and anthropogenic soils, among others. The province also exhibits notable regional climatic diversity. Precipitation demonstrates strong monsoonal characteristics, with abundant rainfall in summer driven by the southeast monsoon, and dry, low-rainfall conditions in winter under the influence of cold, dry northwesterly winds (Figure 1).

2.2. Methodological Framework

Figure 2 presents a flowchart illustrating the current study’s methodology. We constructed a full growth cycle CW model for Cunninghamia lanceolata using measured data acquired from 133 plantation sample plots in Guangdong Province, totaling 16,101 trees. First, DBH and H were selected as predictor variables, and six basic growth functions were compared to identify the optimal base model. Then, five age groups were introduced as dummy variables. Finally, due to the hierarchical structure of our data, we adopted a multilevel nonlinear mixed-effects approach to build a two-level NLME CW model, incorporating forest block- and sample plot-level effects included as random effects.

2.3. Data Collection and Pre-Processing

Chinese Fir Tree Ground Survey Data

Field data for individual trees were collected in March 2024. We were involved in a ground survey conducted across 133 sample plots, each measuring 30 m × 30 m. The ages of the investigated Cunninghamia lanceolata plantations ranged from 2 to 40 years. Data related to dead or fallen trees, dead branches, and understory litter were excluded, along with data affected by natural factors (such as wind damage and pest infestations) and any missing, incorrect, or duplicate measurements. By removing these irrelevant data, we ensured that only healthy and suitable tree samples were included in the model analysis. A total of 16,101 Chinese fir trees were recorded. The collected survey information included species, DBH, H, height to crown base, CW, growth condition, site type, stand age, and age class. All trees in the study area were pure Chinese fir plantations (Table 1). Data collection and processing strictly adhered to standardized field survey protocols, ensuring the accuracy and reliability of the dataset.

2.4. Model Establishment

2.4.1. Variable Selection

Crown width is influenced not only by DBH but also by other stand variables such as tree size, stand vitality factors [41], site conditions [42], and competition among trees [43]. Based on existing research [44,45], we selected the following seven key variables: DBH, H, height to crown base (HCB), stand age (AG), SD, dominant tree height (DH), and canopy closure (CD).
To identify the most relevant growth parameters, Pearson’s correlation analysis was performed on the growth variables, followed by a variance inflation factor (VIF) test (with VIF < 10) [46] to check for multicollinearity and avoid the interdependence of multiple predictor variables. Ultimately, two variables most strongly correlated with crown width were selected as the independent variables for constructing the binary models, dummy variable model, and two-level NLME models. In this study, we randomly divided the data into two datasets, with 70% of the data used for model fitting (11,272 observations) and the other 30% for model validation (4829 observations).

2.4.2. Basic Model Construction

This study developed a crown width growth model for Chinese fir trees based on full growth cycle data collected from individual trees in Guangdong Province. Six biologically meaningful crown width models were selected as candidate base models: the allometric growth model (Model 1), the power function model (Model 2), the exponential function model (Model 3), the logistic model (Model 4), the Richards function model (Model 5), and the linear model (Model 6). The variables with the highest correlation coefficients were chosen as the independent variables to construct a binary model for data fitting.
C W i   = β 0   + β 1 x 1   + β 2   x 2   + β 3   ( x 1 × x 2   ) + ε i  
C W i   = β 0   x 1 β 1 x 2   β 2   + ε i
C W i   = β 0   e x p ( β 1 x 1   + β 2   x 2   ) + ε i
C W i   = β 0   1 + e x p ( − ( β 1 x 1   + β 2   x 2 ) )   + ε i  
C W i   = β 0   ( 1 + e x p ( − ( β 1 x 1   + β 2   x 2 ) ) ) β 3   + ε i
C W i   = β 0     + β 1 x 1 + β 2   x 2   + ε i
where x1 and x2 are the most influential variables with the highest correlation; β1, β2, and β3 are parameters to be estimated; and ε is an error term.
We used four indicators to compare the fitting and predictive performance of the six theoretical crown width growth models: the Coefficient of Determination (R2), Root Mean Squared Error (RMSE), Total Relative Error (TRE), and Akaike Information Criterion (AIC) [47,48]. The model with the best-fitting performance was selected as the base model for constructing the subsequent crown width mixed-effects model.
R 2 = 1 − ∑ i = 1 n ( y i - y i ^ ) 2 / ∑ i = 1 n ( y i - y i ¯ ) 2
R M S E = ∑ i = 1 n ( y i - y i ^ ) 2 / ( n − 1 )
T R E = ∑ y i - y i ^ ∑ y i
A I C = 2 k − 2 ln ( L )
where yi is the observed value of the dependent variable; y i ^ is the predicted value of the dependent variable by the model; y i ¯ is the mean of the observed values; n is the number of observations; k is the number of parameters in the model. L is the maximum likelihood of the model. To evaluate the model’s goodness of fit, statistical metrics such as the Coefficient of Determination (R2), Root Mean Square Error (RMSE), Akaike Information Criterion (AIC), and Total Relative Error (TRE) are used.
Specifically, R2 measures the proportion of variance explained by the model, RMSE quantifies the average prediction error, AIC assesses model fit while considering model complexity, and TRE evaluates the overall relative deviation between observed and predicted values, providing an additional measure of model accuracy.

2.4.3. Construction of Dummy Variable Model

Dummy variables are an essential tool for handling categorical variables, as they convert these variables into binary variables, allowing them to be incorporated into regression models for analysis. The classification of age groups helps distinguish the growth differences in Chinese fir forests at different stages of development. When the model simultaneously considers both the site index effect and the impact of age group on crown width, we first introduce age group (AG) as a dummy variable into the model to evaluate the influence of different stand ages on crown width. The AGs of Cunninghamia lanceolata trees include five categories: young stand (range from 1 to 10 years), middle-aged stand (from 11 to 20 years), near-mature stand (from 21 to 25 years), mature stand (from 26 to 35 years), and over-mature stand are those older than 36 years. By incorporating dummy variables, the model can better illustrate the impact of each developmental stage on the biomass of Chinese fir and analyze the contribution of different age groups to the total biomass.
In model construction, we selected one age group as the reference group and transformed the other age group variables into quantitative variables, assigning values of 0 or 1 in the regression analysis. Generally, when there are n categorical variables, one category is set as the reference group, so the number of dummy variables is n − 1. This approach allows the dummy variable model to flexibly compare the differences between different age groups and the reference group. Not only does this method simplify the handling of age group variables, but it also enhances the model’s interpretative power for multi-category data, enabling a deeper understanding of the stage-specific impact of stand age on crown width growth (Model 11).
A G i = 1   When   for   age   group   i 0   If   not
where AGi is a binary indicator that takes values of 0 or 1.

2.4.4. Two-Level NLME Models Establishment

The model construction was carried out using the “nlme” package in R 4.2.3. The “nlme” package is specifically designed for NLME models, providing the flexibility to handle complex nonlinear models, including nested data structures and nonlinear functional forms. When using NLME models for prediction, it is necessary to estimate the random-effect parameters. In this study, these parameters were derived using partially known information from the sample plots and the restricted maximum likelihood (REML) method (Model 12). The predict function was then employed to generate predictions for the fixed-effects component of the NLME model. Analysis of variance (ANOVA) was used for model comparisons, guiding the selection of the appropriate combination of fixed and random effects [49].
b ^ i = D ^ Z ^ i T ( R ^ i + Z ^ i D ^ Z ^ i T ) − 1 e ^ i
where b ^ i is the estimated random parameter value; D ^ is the q × q random effect variance-covariance matrix (where q is the number of random parameters); R ^ i is the k × k error variance-covariance matrix (where k is the number of error terms); e ^ i is the k × 1 error vector; Z ^ i is the design matrix for the random effect parameters.
The model structure was optimized by comparing indicators such as AIC across different models. During the validation of the two-level NLME Models, the primary focus was on assessing the goodness of fit and the accuracy of the random effects estimates. In evaluating the model’s goodness of fit, residual analysis was first conducted to ensure that the residuals met the assumptions of normality and homogeneity, with no significant heteroscedasticity. When estimating the random effects, the models utilized maximum likelihood method (MLE) to optimize the parameters of both fixed and random effects. This method, by maximizing the likelihood function, accurately estimates the variance and effect size of the random effects, thereby quantifying the variability between different levels.
Based on the dummy variable model, we incorporated forest block- and sample plot-level effects as random effects to develop a nested two-level NLME model for predicting tree crown width. At the first level, block-level effects were treated as a random effect to account for environmental differences and the impact of management practices on tree growth across different blocks. At the second level, sample plot-level effects were included as a random effect to capture variability among plots within the same block. The nested two-level NLME model can more accurately describe complex data structures and elucidate the interactions of factors at different hierarchical levels influencing tree growth.
The fixed effects were used to explain the overall impact of the variables most strongly correlated with crown width, while the two levels of random effects allowed for variability between different blocks and sample plots. This approach not only improved the predictive accuracy of the model but also revealed the growth patterns of trees under different environmental conditions and at various growth stages. Ultimately, the best two-level mixed-effects model was selected to explain and predict the variation in tree crown width. The model was expressed as follows [50]:
y i j = f ( ϕ i , v i j ) + ε i j ,   i = 1 , … , ∑ a = 1 M M a ,   j = 1 , … , n i ,   ϕ i = A i β + B i ( Block ) u i ( Block ) + B i ( Block × Plot ) u i ( Block × Plot ) ,   u i ( 1 ) ~ N ( 0 , Ψ ( Block ) ) ,   u i ( Block × Plot ) ~ N ( 0 , Ψ ( Block × Plot ) )  
where yij represents the crown width value of the j plot in the i block (response variable); ϕi is a linear predictor determined by both fixed effects and random effects; ui represents the random effects; f(ϕi,vij) represents a nonlinear function that depends on the linear predictor ϕi and other covariates vij; ε i j represents the error term, which is usually assumed to follow a normal distribution N ( 0 , σ 2 ) and is used to capture the random error that the model cannot explain; j = 1 , … , ∑ a = 1 M M a represents that there are a total of M blocks, each block containing Ma plots, and the total number of plots is ∑ a = 1 M M a ; j = 1,…,ni represents the number of sample plots in each block; Ai is the design matrix, which includes the predictor variables of the fixed effects; β is the coefficient vector of the fixed effects; B i ( B l o c k ) is the design matrix, which includes the random effect variables related to the interaction with the blocks and sample plots; u i ( B l o c k ) is the random effects at the block level, assumed to follow a multivariate normal distribution with mean 0 and variance N ( 0 , Ψ ( Block ) ) ; B i ( B l o c k × p l o t ) is the design matrix, which includes the random effect variables related to the interaction with the blocks and sample plots; u i ( B l o c k × p l o t ) is the random effect at the interaction level of blocks and sample plots, assumed to follow a normal distribution with a mean of 0 and a variance of N ( 0 , Ψ ( B lock × Plot ) ) . The random effects assumption and the error term assumption are independent of each other and each follows a normal distribution.

3. Results

3.1. Parameter Optimization

Pearson’s correlation tests revealed that seven growth parameters, namely diameter at breast height (DBH), tree height (H), crown base height (HCB), stand age (A), stand density (SD), dominant tree height (DH), and canopy closure (CD), all exhibited strong correlations with CW. As shown in Figure 3a, DBH had the most significant effect on crown width, with a correlation coefficient of 0.69, followed by H, which also showed a significant positive correlation with CW (correlation coefficient of 0.59). Additionally, both DBH and H had VIF values well below 10 (Figure 3b), indicating that there was no significant multicollinearity between these two variables. Therefore, we selected DBH and H as the predictor variables for the model and constructed a binary model with DBH and H as the independent variables.

3.2. Binary Model Results

After determining that DBH and H were the most critical variables influencing tree CW, the aforementioned two factors were utilized as predictor variables in six basic growth models (Model 1–6). The goodness of fit for each model was assessed using various evaluation parameters. Table 2 presents the performance metrics for all six models. Overall, the evaluation indicators were consistent across both the training and validation datasets, with each model achieving an R2 value exceeding 0.47 (Figure 4).
During the basic model selection process, Model 1 exhibited the highest fitting accuracy. However, based on the (approximate) t-test, the parameter d in this model was not statistically significant, rendering the allometric model unsuitable as the optimal base model. Therefore, after a comprehensive evaluation using AIC, RMSE, and TRE, we focused on Model 4 and Model 2, as their fitting accuracies were similar (Model 4’s R2 = 0.475; Model 2’s R2 = 0.471). Ultimately, we chose Model 2, which uses fewer parameters, as the optimal base model. This choice facilitates better model fitting in the subsequent construction of the dummy variable model and the two-level NLME CW model.

3.3. Dummy Variable Model

In Model 2, DBH and H served as core predictor variables, demonstrating strong predictive performance for crown width. Based on previous research findings, we introduced the five age groups of Chinese fir forests (young stands, middle-aged stands, near-mature stands, mature stands, and over-mature stands) as dummy variables into the binary power function model to further evaluate its predictive capability for CW across different developmental stages (Model 14).
Dummy variables were used to represent the different age groups of Chinese fir trees. After individually testing the models, it was found that applying dummy variables to parameter a yielded the best-fitting results. As shown in Table 3, incorporating the qualitative factor of age group improved the model’s R2 to 0.517, representing a 9.77% increment compared to Model 2. Additionally, the AIC decreased by 1.81%, while RMSE and TRE reduced by 2.81% and 5.39%, respectively. These improvements indicated that adding the age group significantly reduced the fitting error and enhanced the overall model performance.

3.4. Prediction Results of Two-Level NLME Models

Based on Model 14, an NLME model (Model 15) incorporating two levels of random effects—the forest block- and sample plot-level effects—was developed. We explored various model configurations and found that the optimal performance was achieved when the block effect was applied to parameters b + c and the sample plot effect was applied to parameter b. Table 4 represents the evaluation metrics for this model.
C W i j k = ( 0.548 A G E 1 + 0.582 A G E 2 + 0.513 A G E 3 + 0.568 A G E 4 + 0.614 A G E 5 ) D B H i j k ( 0.613 + u i j k 1 + v i j k 1 ) H i j k 6.151 + u i j k 2
where CWijk represents the crown width of the k-th tree in the j-th sample plot of the i-th block. AGEi represents dummy variables for the five growth stages, i-th = 1, 2, 3, 4, 5; uik represents the random effect of the k-th tree in the i-th block. vijk represents the random effect of the k-th tree in the j-th sample plot of the i-th block. DBHijk represents the diameter at breast height of the k-th tree in the j-th plot of the i-th block; H represents the height of the k-th tree in the j-th sample plot of the i-th block.
As shown in Table 4, the nested two-level NLME models exhibited a significantly improved predictive performance compared to both the basic bivariate model and the dummy variable model, with an R2 of 0.731. In the modeling results, Model 15 showed a 55.21% improvement in R2 over Model 2. Compared to Model 14, Model 15 achieved a 41.39% increase in R2. Additionally, the RMSE and AIC values were reduced by 27.47% and 27.83% compared to Model 2, and by 25.38% and 26.49%, respectively, compared to Model 14. These results suggested that the nested two-level NLME models, which incorporated the full growth cycle of trees, exhibited significantly greater explanatory power in accounting for the variability of crown width.
The residuals of all three models were centered around zero (Figure 5). However, the residuals of Model 2 and Model 14 exhibited certain heteroscedastic characteristics, with larger fluctuations observed at higher fitted values. In contrast, the residuals of Model 15 were more evenly distributed and showed less heteroscedasticity, indicating its ability to better capture the random effects in the data and reduce heteroscedasticity. Furthermore, in terms of fit accuracy, Model 15 displayed the smallest residual range, outperforming the other models, which further underscores its advantage in handling complex data structures.

4. Discussion

Constructing an individual tree CW model for Cunninghamia lanceolata plantations can significantly reduce measurement time and costs while improving prediction accuracy. This study provided a detailed methodology for developing a CW model by integrating age group dummy variables with a nested two-level NLME approach. The model construction process includes the selection of a base model, identification of predictor variables, development of a dummy variable model, formulation of mixed-effects model parameters, and estimation of random-effect parameters. Ultimately, this study enabled a full growth cycle CW prediction for Chinese fir plantations in southern China. The choice of base model significantly influences the performance of the final mixed-effects model, including its ability to incorporate random effects [17,29,51]. This study, focusing on Cunninghamia lanceolata plantations in Guangdong Province, compared six growth functions and identified the power function model as the optimal base model. This finding agreed with the research of Fu et al. [52], who also concluded that the power function was the most suitable for CW modeling in Chinese fir trees. Although the allometric growth model exhibited the highest predictive accuracy (R2 = 0.490), it contained a large number of variables and parameters compared to the power function model, potentially leading to overparameterization and convergence issues [28,38]. A simple yet reliable model with robust predictive accuracy is ideal for effective forest management [45]. Therefore, we identified the power function model as the most suitable basic function for constructing a two-level NLME CW model. Then, various tree variables were examined for their contributions to improving the CW model’s fit. Based on Pearson’s correlation coefficients, DBH and H had the strongest influence on CW, leading us to select DBH as the core predictor in the binary model, with H as the secondary independent variable. This selection was consistent with previous studies [52,53]. Changes in DBH and H are two fundamental aspects of tree growth [54], reflecting a balance between horizontal and vertical expansion [55]. Using DBH as the core factor in the CW model aligned with previous research findings [2,56,57,58]. The allometric relationship between DBH and CW has been well documented across various tree species [2,59]. In dynamic growth prediction, DBH has been recognized as one of the most reliable indicators for estimating CW, overall tree growth status, and timber volume due to its ease of measurement and strong correlation with growth [60,61]. Meanwhile, H serves as a key factor influencing CW variability. It not only represents the overall growth status of a tree but also captures additional information on CW variation that DBH alone cannot fully explain [62]. For example, in different age groups, the contribution of H to CW prediction is particularly significant [63].
To establish a full growth cycle CW model for trees, we introduced age group dummy variables to quantify the stage-specific effects of different age groups on Cunninghamia lanceolata crown growth. This approach effectively addressed the challenge of incorporating categorical variables into regression models. The results demonstrated that incorporating dummy variables led to substantial model improvements. Compared to the optimal basic model, the dummy variable model’s R2 increased from 0.483 to 0.517, RMSE decreased from 0.677 to 0.658, and TRE reduced from 6.626 to 6.269. The inclusion of dummy variables allowed for a more precise representation of CW variations across different growth stages, avoiding the oversimplification inherent in traditional regression methods. Consequently, the model’s applicability across the entire growth cycle was significantly enhanced. This refinement not only reinforced the importance of age group classification in CW modeling but also improved the model’s ecological interpretability. Previous studies have primarily focused on analyzing CW characteristics within a specific growth stage or under static conditions [11,17,40]. A number of studies have explored the correlations and path analysis between DBH, H, and CW across different age groups and the structural changes in CW with increasing stand age [31,32]. Some relevant research has investigated differences in aboveground biomass allocation and growth rates at various age stages [34,64,65]. However, relatively few studies have incorporated a full growth cycle perspective into CW prediction using mixed-effects modeling. In contrast, the model developed in this present study provided a more comprehensive representation of the long-term CW growth trajectory, resulting in greater predictive accuracy and practical applicability. This advancement offered valuable scientific support for forest resource monitoring, ecological conservation, and sustainable management.
When constructing a full growth cycle CW prediction model, we found that introducing dummy variables to quantify the effects of different age groups on CW growth could capture stage-specific dynamic changes to some extent. However, this approach resulted in only a limited improvement in model accuracy, with R2 increasing by just 0.034 compared to the power function model. In previous forest growth and yield models, traditional regression methods were commonly used to establish linear relationships between CW and various stand factors. These methods generally assume that the dataset is independent and homogeneous [66]. In practice, CW data are often derived from repeated measurements or hierarchical data structures [38]. Researchers frequently conduct repeated observations of tree CW within the same plots under different site conditions or observe the same trees multiple times over different time periods [44,45]. This violates the assumption of data independence and leads to significant autocorrelation. Additionally, regional environmental variations and plot heterogeneity influence tree growth processes, and failing to account for these factors can result in biased descriptions of CW growth dynamics [8,67,68]. In recent years, mixed-effects modeling has been widely applied as it effectively addresses such issues, providing more reliable predictions and improving model robustness in forestry research [48]. For instance, Fu et al. [52], Zhong et al. [51], Nie et al. [69], Wang et al. [70], and others found that incorporating random effects significantly improved model accuracy when using NLME models to predict CW. Due to the hierarchical structure of our data, we further used the forest block- and sample plot-level effects included as random effects in the NLME model. The results demonstrated an additional improvement in predictive accuracy over the dummy variable model. Specifically, the two-level NLME exhibited the highest accuracy, with R2 reaching 0.731, TRE decreasing to 3.373, and RMSE decreasing to 0.491. Compared to the power function model and the dummy variable model, the two-level NLME model achieved an increase in R2 by 0.242 and 0.214, a reduction in RMSE by 0.186 and 0.027, and a decrease in TRE by 3.253 and 2.896, respectively. These findings indicated that incorporating random effects can significantly enhance model performance. This conclusion was consistent with other studies [28,38,51], confirming that mixed-effects modeling provides greater flexibility in accounting for heterogeneity and randomness arising from both known and unknown factors. According to Zhong et al.’s study on Cunninghamia lanceolata plantations in Jiangxi Province, a mixed-effects model incorporating plot-level random effects outperformed the base generalized model, with R2 increasing by 27.86%, ERMS decreasing by 35.73%, and EMA decreasing by 51.87% [51]. Similarly, Zhou et al. developed a two-level NLME CW model for Phyllostachys edulis (Moso bamboo), incorporating both the block- and plot-level effects as random factors [38]. Their model achieved the best-fitting statistics, characterized by the highest R2 and the lowest RMSE and TRE. Furthermore, our study found that the nested two-level NLME model based on the power function successfully removed some of the heteroscedasticity (as evidenced by the residual distribution of the model fitting dataset in Figure 5). The model effectively addressed the nested data structure resulting from observational dependencies.
Our findings indicated that incorporating the forest block- and plot-level effects as random factors in the NLME model effectively accounts for CW variations driven by environmental heterogeneity. Specifically, the block effect captures regional environmental variations, such as climate conditions and soil types, which exert long-term influences on CW growth [71]. In contrast, the plot effect reflects tree-to-tree competition and resource allocation patterns at a micro-scale [17]. The final full growth cycle CW prediction model, developed using the NLME approach, not only accounted for the nested effects of block and plot levels, successfully disentangling regional (forest block) and plot-level environmental heterogeneity, but also demonstrated the critical role of incorporating tree age in NLME models. This incorporation is essential for enhancing model efficiency and predictive accuracy, ultimately enabling accurate full growth cycle CW predictions for Cunninghamia lanceolata.
From the above analysis, we utilized a comprehensive dataset to develop a full growth cycle CW model, revealing the dynamic characteristics of CW growth and its multi-scale influencing mechanisms in Cunninghamia lanceolata plantations. We recommend that when predicting CW in Cunninghamia lanceolata plantations, in addition to analyzing the random effects introduced by blocks and the nested plots within them, it is also essential to account for the CW variations across different age stages. This approach can further enhance model predictive accuracy and provide a more robust representation of CW growth dynamics.

5. Conclusions

This study developed a comprehensive full growth cycle CW model to estimate the CW of individual Chinese fir trees in pure plantation stands in southern China, integrating power function, dummy variable, and two-level NLME models to reveal the dynamic growth patterns and multi-scale mechanisms influencing crown development. Based on statistical results produced from fitting the six basic growth functions, we identified the most suitable function (the power function) to construct a two-level NLME CW model. The results showed that the basic model incorporating DBH and H as key predictors effectively captured the primary relationship between CW and tree growth (Model 2’s R2 = 0.483, RMSE = 0.677). Merely introducing dummy variables to account for age group effects only partially captured the stage-specific dynamics, resulting in limited improvements (Model 14’s R2 = 0.517, RMSE = 0.658) in model accuracy. Our dummy variable model still exhibited autocorrelation and heteroscedasticity. Recognizing that regional environmental differences and plot heterogeneity significantly influence tree growth, this study adopted a multilevel nonlinear mixed-effects approach. Some of the heteroscedasticity was successfully reduced by the two-level NLME model. The prediction accuracy of the NLME model with forest block- and sample plot-level effects included as random effects was substantially enhanced (Model 15’s R2 = 0.731, RMSE = 0.491), suggesting the necessity of considering environmental heterogeneity and random effects in CW prediction. Overall, this research not only offers a robust framework for modeling crown width throughout the full growth cycle of Chinese fir trees but also provides practical guidance for sustainable forest management.

Author Contributions

Z.W.: Conceptualization, Methodology, Formal analysis and investigation, Writing—original draft preparation, Visualization. D.X.: Formal analysis and investigation, Software. Z.L.: Methodology, Software. L.F. (Linyang Feng): Visualization, Supervision, Writing—review. Q.Y.: Methodology, Visualization, Writing—review and editing. J.Y.: Software. Q.W.: Visualization, Supervision. X.L.: Software. Y.W.: Visualization. R.P.S.: Writing—review and editing. L.F. (Liyong Fu): Conceptualization, Project administration, Funding acquisition, Writing—review and editing. All authors have read and agreed to the published version of the manuscript.

Funding

This study is supported by the Fundamental Research Funds for the Central Nonprofit Research Institution of CAF under Grant CAFYBB2022ZB002.

Institutional Review Board Statement

Not applicable.

Informed Consent Statement

Not applicable.

Data Availability Statement

Data are contained within the article.

Conflicts of Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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Figure 1. Sample point distribution map.
Figure 1. Sample point distribution map.
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Figure 2. Methodology framework.
Figure 2. Methodology framework.
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Figure 3. Pearson’s correlation coefficients and collinearity tests among multiple variables and crown width: (a) correlation coefficients of CW with other variables; (b) collinearity of CW with each variable, where the y−axis shows the VIF values. CW, crown width; A, stand age (year); SD, stand density; CD, canopy closure; DBH, diameter at breast height; H, tree height; HCB, height to crown base; DH, dominant tree height.
Figure 3. Pearson’s correlation coefficients and collinearity tests among multiple variables and crown width: (a) correlation coefficients of CW with other variables; (b) collinearity of CW with each variable, where the y−axis shows the VIF values. CW, crown width; A, stand age (year); SD, stand density; CD, canopy closure; DBH, diameter at breast height; H, tree height; HCB, height to crown base; DH, dominant tree height.
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Figure 4. Comparison of model performance for predicting Chinese fir crown width (RMSE, Root Mean Square Error; TRE, Total Relative Error; blue points, training dataset; orange points, validation dataset).
Figure 4. Comparison of model performance for predicting Chinese fir crown width (RMSE, Root Mean Square Error; TRE, Total Relative Error; blue points, training dataset; orange points, validation dataset).
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Figure 5. Distribution of residuals for three models predicting the CW of Chinese fir trees.
Figure 5. Distribution of residuals for three models predicting the CW of Chinese fir trees.
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Table 1. Summary statistics for model fitting and validation data.
Table 1. Summary statistics for model fitting and validation data.
DATAVariableMinMaxMeanStd
Fitting dataCW (m)0.27.052.5410.93
A (year)24018.7210.33
SD (trees ha−1)0.03216.592.53
CD (%)650653339761385.88
DBH (cm)2.449.914.366.07
H (m)0.927.511.154.23
HCB (m)0.1216.0612.89
DH (m)82716.594.19
Validation dataCW (m)0.557.052.5410.95
A (year)64018.6810.48
SD (trees ha−1)0.0321.0292.63
CD (%)650653339911398.69
DBH (cm)2.746.114.356.07
H (m)1.631.511.154.23
HCB (m)0.3226.0933.46
DH (m)82716.524.29
Min, minimum; Max, maximum; Mean, average value; Std, standard deviation; CW, crown width; A, stand age (year); SD, stand density; CD, canopy closure; DBH, diameter at breast height; H, tree height; HCB, height to crown base; DH, dominant tree height.
Table 2. Binary model fitting accuracy.
Table 2. Binary model fitting accuracy.
Model ParametersFitting DataValidation Data
Model FormabcdeAICR2RMSETRER2RMSETRE
Model10.9210.1210.978−0.0018.58423,1400.4780.6756.63560.4900.6766.655
SE0.0570.0150.0340.0011.167
Model20.5210.612−0.01 23,2890.4710.6796.7330.4830.6776.626
SE0.0090.0120.013
Model31.4840.0320.005 23,7290.4490.6937.0190.4780.6836.829
SE0.0110.0010.001
Model45.4833.7610.086−0.005 23,2120.4750.6776.6820.4870.6716.551
SE0.1320.0830.0030.002
Model54.9510.0510.003 23,6960.4510.6926.9970.4740.6876.891
SE0.0730.0020.001
Model60.108−0.0031.027 23,2340.4730.6786.6980.4870.6716.558
SE0.0020.0030.018
a, b, c, d, e, the model parameters; AIC, Akaike Information Criterion; R2, Coefficient Of Determination; RMSE, Root Mean Square Error; TRE, Total Relative Error.
Table 3. Dummy variable model fitting accuracy.
Table 3. Dummy variable model fitting accuracy.
Model Form C W = ( a 1 A G E 1 + a 2 A G E 2 + a 3 A G E 3 + a 4 A G E 4 + a 5 A G E 5 ) + D B H b + H c (14)
Model ParametersFitting DataValidation Data
Model 14a1a2a3a4a5bcAICR2RMSETRER2RMSETRE
0.5480.5820.5130.5680.6140.613−0.04622,8670.4910.6676.4640.5170.6586.269
SE0.0110.0120.0110.0130.0140.0120.013
a, b, c, the model parameters; AIC, Akaike Information Criterion; R2, Coefficient Of Determination; RMSE, Root Mean Square Error; TRE, Total Relative Error.
Table 4. Fitting results of the two-level NLME models.
Table 4. Fitting results of the two-level NLME models.
Model ParametersFitting DataValidation Data
Model formAICR2RMSETRER2RMSETRE
Model 1416,8080.7170.4973.4890.7310.4913.373
SEa1: 0.011a2: 0.019a3: 0.019a4: 0.018a5: 0.017b: 0.035c: 0.045
a, b, c, the model parameters; AIC, Akaike Information Criterion; R2, Coefficient Of Determination; RMSE, Root Mean Square Error; TRE, Total Relative Error.
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MDPI and ACS Style

Wu, Z.; Xie, D.; Liu, Z.; Feng, L.; Ye, Q.; Ye, J.; Wang, Q.; Liao, X.; Wang, Y.; Sharma, R.P.; et al. Development of Full Growth Cycle Crown Width Models for Chinese Fir (Cunninghamia lanceolata) in Southern China. Forests 2025, 16, 353. https://doi.org/10.3390/f16020353

AMA Style

Wu Z, Xie D, Liu Z, Feng L, Ye Q, Ye J, Wang Q, Liao X, Wang Y, Sharma RP, et al. Development of Full Growth Cycle Crown Width Models for Chinese Fir (Cunninghamia lanceolata) in Southern China. Forests. 2025; 16(2):353. https://doi.org/10.3390/f16020353

Chicago/Turabian Style

Wu, Zheyuan, Dongbo Xie, Ziyang Liu, Linyan Feng, Qiaolin Ye, Jinsheng Ye, Qiulai Wang, Xingyong Liao, Yongjun Wang, Ram P. Sharma, and et al. 2025. "Development of Full Growth Cycle Crown Width Models for Chinese Fir (Cunninghamia lanceolata) in Southern China" Forests 16, no. 2: 353. https://doi.org/10.3390/f16020353

APA Style

Wu, Z., Xie, D., Liu, Z., Feng, L., Ye, Q., Ye, J., Wang, Q., Liao, X., Wang, Y., Sharma, R. P., & Fu, L. (2025). Development of Full Growth Cycle Crown Width Models for Chinese Fir (Cunninghamia lanceolata) in Southern China. Forests, 16(2), 353. https://doi.org/10.3390/f16020353

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