Predicting the Stand Growth and Yield of Mixed Chinese Fir Forests Based on Their Site Quality, Stand Density, and Species Composition

Abstract

The Chinese fir (Cunninghamia lanceolata) is the largest tree species used for afforestation in China. The purpose of this study was to explore the effects of site quality, stand density, and tree species composition on the growth and yield of mixed Chinese fir forests and to build prediction models for their stand average DBH (diameter at breast height), average height, and volume. Using 430 plots of mixed Chinese fir forests in the Fujian Province of China, the optimal base models for predicting stand average DBH, average height, and volume were selected from the Schumacher, Korf, Logistic, Mitscherlich, and Richards equations. On this basis, the site class index (SCI), stand density index (SDI), and tree species composition coefficient (TSCC) were introduced to improve the model’s performance, and the applicability of the different models was evaluated. The optimal base models for the average DBH, average height, and stand volume of mixed Chinese fir forests all used the Richards equation. The best fitting effect was obtained when the SCI was introduced into parameter a in the average height model, while the inclusion of the TSCC did not improve the model significantly. The fitting effects of the average DBH and stand volume models were both best in the form of $y=a_{1}SC{I}^{a_2}[1-\exp (-b_{1}SD{I}^{b_2})t{]}^c$ when the SCI and SDI were introduced. When the TSCC was further included, the fitting effects of the stand average DBH and volume models were significantly improved, with their R² increased by 47.47% and 58.45%, respectively, compared to the base models. The optimal models developed in this study showed good applicability; the residuals were small and distributed uniformly. We found that the SCI had an impact on the maximum values of the stand average DBH, average height, and volume; the SDI was closely related to the growth rate of the diameter and volume, while the TSCC influenced the maximum values of the stand average DBH and volume. The model system established in this study can provide a reference for the harvest prediction and mixing ratio optimization of mixed Chinese fir forests.

1. Introduction

Chinese fir (Cunninghamia lanceolata) is a fast-growing tree species unique to China, with a wide distribution area and high economic value [1,2,3]. According to the Ninth Chinese National Forest Resources Inventory [4], Chinese fir plantations have reached 9.9 million hectares, making them the most widespread plantation tree species in China. With the advantages of good texture and a high yield, the Chinese fir plays an important role in the country’s wood supply and economic development [5,6]. The wood of the Chinese fir is widely used in decoration, construction, shipbuilding, bridge construction, and other fields, and the root and bark of the Chinese fir have certain medicinal values. The growth period of the Chinese fir is quite short, but the economy around its cultivation is considerable. Therefore, Chinese fir plantations have long been a hot research topic among forestry researchers [1,2,3,5,6,7,8,9,10].

As the trend of forest management develops worldwide, the mixed forest performs the functions of enhancing forest stability, increasing species diversity, improving site quality, and enhancing protective benefits [11,12,13]. In particular, the conifer–broadleaf mixed forest has a higher productivity, ecological resilience, and resistance to pests and diseases [14,15,16]. Under traditional intensive management methods, the ecosystem structure of pure Chinese fir forests is quite simple, and problems of soil erosion and productivity decline have appeared repeatedly [15]. Constructing a mixed forest is an effective way to overcome the fertility decline in pure Chinese fir forests, as it can effectively enrich the tree species’ structure, improve nutrient cycling, and improve the forest’s ability to resist natural disasters.

Chinese fir can be mixed with coniferous species as well as broad-leaved species. The main conifer used for mixing with Chinese fir is Pinus massoniana, while the broad-leaved species used for mixing are diverse and include Schima superba, Castanopsis hystrix, Michelia macclurei, and Phoebe bournei. Research has shown that different mixed tree species and mixing ratios have different effects on the growth of Chinese fir [17,18]. Therefore, there is an urgent need for deeper exploration and research on the construction of mixed forests with Chinese fir, the selection of the mixed species and the mixing ratio, and the prediction of stand growth and yield.

The stand growth and yield models are important tools for forest management [19]. They describe the relationship between tree growth, the stand state, and site conditions in the form of one or a set of mathematical equations, enabling the prediction of stand growth and harvest yields [20]. Most stand-level growth and yield models are used for pure forests, using age to predict the average DBH (diameter at breast height), average height, and volume of the stand. Many studies have shown that stand height increases with the improvement in the site quality, while the average DBH and stocking volume are closely related to the site quality and stand density [21,22,23]. Generally, the average DBH and stand volume also increase with high site quality. A stand with a larger density had a smaller average DBH, while a stand with a lower density had a larger average DBH. The impact of stand density on volume occurred when, after reaching a certain density range, stand volume increased with the increase in stand density [24].

At present, many growth and yield models for pure Chinese fir forests have been developed, but there are few studies on mixed Chinese fir forests, and a prediction model combining site quality, stand density, and tree species composition is still lacking [25,26]. Therefore, the purpose of this study is to build growth and yield models for mixed Chinese fir forests, considering site quality, stand density, and tree species composition. The aim is to provide a reference for accurately predicting the yield, optimizing management plans, and evaluating the forest assets of mixed Chinese fir forests.

2. Materials and Methods

2.1. Study Area

The study area is located in the Fujian Province of southeast China (23°33′~28°20′ N, 115°50′~120°40′ E). Fujian Province has a subtropical climate with abundant rainfall and sufficient sunlight. The annual average temperature is 17–21 °C, and the average rainfall is 1400–2000 mm. The accumulated temperature of 10 °C or higher in 70% of Fujian Province reaches 5000–7600 °C. Fujian is the province with the highest forest coverage rate in China, with an area of 8.12 million hectares and a coverage rate of 66.8% [4]. The main tree species in Fujian Province include Cunninghamia lanceolata, Pinus massoniana, Phyllostachys edulis, Schima superba, Cinnamomum camphora, and so on.

2.2. Data

Data in this study were collected from the re-measured permanent plots in the National Forest Inventory (NFI) and the Forest Management and Planning Inventory (FMPI) of Fujian Province. The NFI plots were measured in 2008, 2013, and 2018, and the FMPI plots were measured in 2007 and 2017. All plots were distributed in mountainous areas and some coastal cities in Fujian Province, and each plot was square with an area of 0.067 ha. As a result, a total of 430 plots were generated. Figure 1 displays the locations of the sampling plots.

The tree species mixed with Chinese fir were mainly Pinus massoniana and broad-leaved species, including Schima superba, Phoebe bournei, Cinnamomum camphora, Liquidambar formosana, Quercus glauca, Acacia melanoxylon, Acacia concinnatai, and Castanopsis hystrix. After removing abnormal data using the threefold standard deviation method, 418 plots remained, of which two-thirds (278 plots) were randomly selected for model fitting and one-third (140 plots) was used for model validation.

Table 1. Statistics of stand attributes for fit and validation data.

Data Type	Stand Attributes	Mean	Min.	Max.	SD	CV%
Fit data (278 plots)	Average DBH (cm)	13.83	6.50	24.30	3.60	26.06
	Average height (m)	11.09	4.20	19.00	3.24	29.24
	Stand age (a)	22.96	6.00	47.00	7.74	33.69
	Stand volume (m3/ha)	153.75	11.80	421.46	60.84	39.57
	Stand density (trees/ha)	1918.55	345.00	4560.00	914.88	47.69
Validation data (140 plots)	Average DBH (cm)	14.60	7.80	24.60	3.65	25.02
	Average height (m)	11.59	5.00	19.20	2.99	25.78
	Stand age (a)	23.70	5.00	44.00	8.01	33.80
	Stand volume (m3/ha)	153.86	17.34	414.83	64.88	42.17
	Stand density (trees/ha)	1783.71	330.00	4590.00	936.94	52.53

displays the stand attribute statistics for both fit and validation data.

2.3. Methods

2.3.1. Base Model

Five common theoretical growth equations, namely, the Schumacher, Korf, Logistic, Mitscherlich, and Richards equations, were selected as the base models. The model forms are as follows:

$$\mathrm{Schumacher} (\mathrm{M}1): y=a\exp (-b/t)$$

(1)

$$\mathrm{Korf} (\mathrm{M}2): y=a\exp (-b{t}^{-c})$$

(2)

$$\mathrm{Logistic} (\mathrm{M}3): y=a/[1+b\exp (-ct)]$$

(3)

$$\mathrm{Mitscherlich} (\mathrm{M}4): y=a[1-\exp (-bt)]$$

(4)

$$\mathrm{Richards} (\mathrm{M}5): y=a{[1-\exp (-bt)]}^c$$

(5)

where y represents the dependent variable of stand factors, including stand average DBH (D), average height (H), and volume (V); t represents stand age; and a, b, and c are model parameters.

2.3.2. Site Quality

In this study, the site class index (SCI) represented stand site quality, which is a quantitative indicator that evaluates forest site quality based on the average height of the dominant tree species at reference age. Compared to other site indicators, the high accessibility of the stand average height gives the SCI an advantage in modeling when using large-scale forest inventory data [27]. The SCI takes the prediction model of the stand average height as the guide curve and uses the change in stand age to describe the change in stand average height. The stand average height model was also selected from the five base models (Equations (1)–(5)).

When the SCI was introduced into the base models, their parameters were re-parameterized in exponential form. For example, parameters a, b, and c were set as $a_{1}SC{I}^{a_2}$, $b_{1}SC{I}^{b_2}$, and $c_{1}SC{I}^{c_2}$, respectively.

2.3.3. Stand Density

The stand density index (SDI), which is calculated by converting the number of trees in a real stand to the number of trees per unit area when the stand has a standard average diameter (also known as comparison diameter), was used to describe stand density. The SDI can not only reflect the crowding degree of trees in the stand, but also has no close relationship with stand age and site conditions [28]. Therefore, the SDI is widely used in forest modeling and management practices [29].

The calculation of the SDI is as follows:

$$SDI=N(D_0/D{)}^b$$

(6)

where N is the number of trees per hectare; D₀ is the standard average diameter, which is 20 cm in this study; D is the actual stand average diameter; and b is the slope of the maximum density line, which was calculated as −1.4913 in this study.

Similar to the SCI, the parameters of the base models were also re-parameterized in the exponential form of the SDI to include stand density in the models. For example, parameters a, b, and c were set as $a_{1}SD{I}^{a_2}$, $b_{1}SD{I}^{b_2}$, and $c_{1}SD{I}^{c_2}$, respectively.

2.3.4. Tree Species Composition

The tree species composition coefficient (TSCC) was used to describe tree species composition in Chinese fir mixed forests. The TSCC was calculated as the proportion of the volume of each tree species to the total volume of the stand [30]. In this study, the tree species in Chinese fir mixed forests were divided into three species groups: Cunninghamia lanceolata, Pinus massoniana, and broad-leaved species. The value of the TSCC for each tree species ranges from 0 to 1, and the sum of the TSCC for all tree species in a stand is 1. We used L₁, L_2, and L₃ to represent the TSCC of Cunninghamia lanceolata, Pinus massoniana, and broad-leaved species, respectively; thus, L₁ + L₂ + L₃ = 1.

In this study, the TSCC was introduced into the models after the SCI and SDI were already considered, and the model parameters were re-parameterized in the linear form of the TSCC for all three species groups. For example, parameter a₁ was set as (a₁₁L₁ + a₁₂L₂ + a₁₃L₃).

2.3.5. Model Evaluation Statistics

The root mean square error (RMSE), mean absolute relative error (MARE), and coefficient of determination (R²) were applied to evaluate the different models. The smaller the values of RMSE and MARE and the larger the value of R², the better the fitting effect of the model. Meanwhile, the total relative error (TRE), mean relative error (MRE), MARE, and prediction accuracy (P) were used as evaluation criteria to test the applicability of the model. The formula of each index is as follows:

$$RMSE=\sqrt{\left({\sum }_{i=1}^n(X_i-X_i^{\prime }{)}^2\right)/n}$$

(7)

$$MARE=\frac{1}{n}{\sum }_{i=1}^n\left|(X_i-X_i^{\prime })/{X_i}^{\prime }\right|\times 100\%$$

(8)

$$R^2=1-{\sum }_{i=1}^n(X_i-X_i^{\prime }{)}^2/{\sum }_{i=1}^n(X_i-\overline{X_i}{)}^2$$

(9)

$$TRE=({\sum }_{i=1}^n{X}_i-{\sum }_{i=1}^n{X_i}^{\prime })/{\sum }_{i=1}^n{X_i}^{\prime }\times 100\%$$

(10)

$$MRE=\frac{1}{n}{\sum }_{i=1}^n((X_i-X_i^{\prime })/X_i^{\prime })\times 100\%$$

(11)

$$P=\left[1-t_{0.05}\sqrt{{\sum }_{i=1}^n(X_i-X_i^{\prime }{)}^2}/(\overline{X_i^{\prime }}\sqrt{n(n-k)})\right]\times 100\%$$

(12)

where X_i and $X_i^{\prime }$ are respectively the observed and estimated values of D, H, or M; $\overline{X_i}$ and $\overline{X_i^{\prime }}$ are respectively the average values of X_i and $X_i^{\prime }$; n is the number of plots, k is the number of parameters, and t_0.05 represents the t distribution value at the confidence level α = 0.05.

3. Results

3.1. Selection of Base Model

The prediction models for stand average DBH (D), average height (H), and stand volume (V) were developed utilizing the five candidate base models (Equations (1)–(5)). The model parameters and evaluation statistics were calculated to select the optimal base model for each of the dependent variables. From

Table 2. Parameter estimates and fitting statistics of the base models.

Dependent Variable	Model No.	Model	Parameter Estimation			Fitting Statistic
			a	b	c	RMSE	MARE%	R2
D (cm)	M1	Schumacher	28.2527	11.0819		2.976	19.37	0.56
	M2	Korf	25.2272	23.9707	1.3224	2.905	18.90	0.59
	M3	Logistic	21.4882	7.1317	0.1572	2.844	18.50	0.61
	M4	Mitscherlich	24.0049	0.0536		2.879	18.73	0.60
	M5	Richards	22.7565	0.0912	2.0471	2.811	18.29	0.63
H (m)	M1	Schumacher	17.5724	11.3259		2.824	22.93	0.49
	M2	Korf	12.8521	473.5443	2.5784	2.756	22.37	0.53
	M3	Logistic	12.0700	19.5583	0.2446	2.757	22.38	0.53
	M4	Mitscherlich	14.5786	0.0572		2.845	23.10	0.48
	M5	Richards	17.2321	0.1079	1.9213	2.587	21.00	0.60
V (m3/ha)	M1	Schumacher	259.2937	11.9968		52.390	23.85	0.51
	M2	Korf	182.5759	1192.0277	2.9076	52.720	24.00	0.50
	M3	Logistic	176.6615	32.5480	0.2728	50.629	23.05	0.55
	M4	Mitscherlich	213.5241	0.0544		52.838	24.06	0.50
	M5	Richards	338.0843	0.0502	1.9039	48.329	22.00	0.61

Note: Bold numbers denote the best model for each criterion.

, the Richards equation (M5) exhibited the highest R² and the lowest RMSE and MARE for all the prediction models of Chinese fir mixed forests. The R² values of the Richards equation for predicting D, H, and V were 0.63, 0.60, and 0.61, RMSE values were 2.811, 2.587, and 48.329, and MARE values were 18.29%, 21.00% and 22.00%, respectively. Therefore, the Richards equation (M5) was selected as the optimal base model for predicting average DBH, average height, and stand volume of Chinese fir mixed forests. The three optimal base models are as follows:

$$D=22.7565[1-\exp (-0.0912t){]}^{2.0471}$$

(13)

$$H=17.2321[1-\exp (-0.1079t){]}^{1.9213}$$

(14)

$$V=338.0843[1-\exp (-0.0502t){]}^{1.9039}$$

(15)

3.2. Height Prediction Model Based on the SCI

Since the Richards equation was the optimal model for stand average height, it was also used to construct the site class index model. Using Equation (14) as the guide curve, the SCI is the stand average height at the reference age t₀. The SCI is calculated as follows:

$$SCI=H(\frac{1-e^{-b{t}_0}}{1-e^{-bt}}{)}^c$$

(16)

where H is stand average height; t is stand age; t₀ is the reference age, which is 20 in this study; a, b, c are the parameters in Equation (14), i.e., a = 17.2321, b = 0.1079, c = 1.9213.

On the basis of the Richards equation, the stand average height model including the SCI was constructed by introducing the SCI into all parameters of the equation. Parameters a, b, and c were re-parameterized in the exponential form of the SCI, and they were set as $a_{1}SC{I}^{a_2}$, $b_{1}SC{I}^{b_2}$, and $c_{1}SC{I}^{c_2}$, respectively. The estimated parameters and fitting statistics for each model are shown in

Table 3. Parameter estimates and fitting statistics of the average height models containing SCI.

Model No.	Parameterization	Parameter Estimation									Fitting Statistic
	SCI	a	a1	a2	b	b1	b2	c	c1	c2	RMSE	MARE%	R2
M6	a		1.3888	0.9788	0.1028			1.9043			0.823	7.93	0.97
M7	b	14.7145				0.0008	1.8621	1.0951			1.723	15.54	0.85
M8	c	1204.5296			9.6762				0.5910	−0.2036	1.971	17.78	0.79
M9	a, b		1.0833	1.0210		0.2145	−0.0037	8.8912			0.938	8.46	0.96
M10	a, c		1.0887	1.0188	0.2127				8.8992	0.0002	0.938	8.46	0.96
M11	b, c	14.2879				0.0002	2.6097		0.0307	1.6788	1.635	14.75	0.86
M12	a, b, c		1.0522	1.0329		0.2736	−0.1028		17.4751	−0.2742	0.938	8.46	0.96

Note: Bold numbers denote the best model for each criterion. a, b and c are the parameters of the base model M5 for height prediction. For M6, parameter a is re-parameterized to $a_{1}SC{I}^{a_2}$ by the SCI; thus, its equation becomes $H=a_{1}SC{I}^{a_2}{[1-\exp (-bt)]}^c$. The other equations can be deduced by analogy.

.

Compared to the base model, the fitting results of the seven models considering the SCI were greatly improved, with the best performance being model M6, which introduced the SCI into parameter a, which received the largest R² and the smallest RMSE and MARE values. The R² of M6 was 0.36 larger than that of the base model, and the RMSE and MARE were 1.763 and 13.07% lower than that of the base model, respectively. Following M6 were models M9, M10, and M12, which introduced the SCI into parameters a and b, a and c, and a, b, and c, respectively. The common denominator of these four models was the introduction of the SCI into parameter a, and the fitting effects were similar. In order to simplify the model and avoid overfitting, M6 was chosen as the prediction model for stand average height, and its expression is:

$$H=(1.3888SC{I}^{0.9788}){[1-\exp (-0.1028t)]}^{1.9043}$$

(17)

3.3. Diameter and Volume Prediction Model Based on the SCI and SDI

The stand average DBH and volume models containing the SCI and SDI were developed on the basis of their optimal base model (Equations (13) and (15)). Parameters a, b, and c were re-parameterized by the SCI and SDI in exponential form and were set as $a_{1}SC{I}^{a_2}$, $b_{1}SC{I}^{b_2}$, $c_{1}SC{I}^{c_2}$, $a_{1}SD{I}^{a_2}$, $b_{1}SD{I}^{b_2}$, and $c_{1}SD{I}^{c_2}$, respectively.

3.3.1. Stand Prediction Model of Average DBH

The parameter estimates and fitting statistics of the diameter prediction models containing the SCI and SDI are shown in

Table 4. Parameter estimates and fitting statistics of the average DBH models containing the SCI and SDI.

Model No.	Parameterization		Parameter Estimation									Fitting Statistic
	SCI	SDI	a	a1	a2	b	b1	b2	c	c1	c2	RMSE	MARE%	R2
M13	a	b		20.3737	0.0353		0.0908	0.0181	2.2214			2.005	13.05	0.83
M14	a	c		20.3460	0.0359	0.1032				2.6084	−0.0225	2.013	13.10	0.83
M15	b	a		20.7651	0.0090		0.0902	0.0585	2.2396			2.065	13.44	0.82
M16	b	c	22.1461				0.0896	0.0588		2.5884	−0.0221	2.066	13.44	0.82
M17	c	a		20.7387	0.0092	0.1035				2.7449	−0.0868	2.066	13.45	0.82
M18	c	b	22.1481				0.0903	0.0182		2.6968	−0.0850	2.066	13.44	0.82
M19	a	b, c		20.4317	0.0330		0.0412	0.1338		0.4726	0.2272	2.016	13.12	0.83
M20	b	a, c		20.7263	0.0093		0.0903	0.0585		2.2249	0.0010	2.065	13.44	0.82
M21	c	a, b		22.7224	−0.0036		0.0869	0.0235		2.6843	−0.0846	2.065	13.44	0.82
M22	a, b	c		21.5970	0.0106		0.0928	0.0443		2.5939	−0.0222	2.013	13.10	0.83
M23	b, c	a		20.8502	0.0084		0.0499	0.3128		0.7119	0.4923	2.066	13.45	0.82
M24	a, c	b		20.5202	0.0323		0.0907	0.0181		2.2784	−0.0110	2.012	13.09	0.83

Note: Bold numbers denote the best model for each criterion. a, b, and c are the parameters of the base model M5 for diameter prediction. For M13, parameters a and b are re-parameterized to $a_{1}SC{I}^{a_2}$ and $b_{1}SD{I}^{b_2}$ by the SCI and SDI, respectively; thus, its equation becomes $D=a_{1}SC{I}^{a_2}{[1-\exp (-b_{1}SD{I}^{b_2}t)]}^c$. The other equations can be deduced by analogy.

. The fitting effects of different models were relatively close, and all fitting criteria performed well and showed significant improvement over the base model. Among them, the model with the best performance was M13 in which the SCI and SDI were introduced into parameters a and b, respectively. The R² of M13 increased by 32.77% compared with the base model, while the RMSE and MARE decreased by 0.806 and 5.24%, respectively. Therefore, M13 was selected as the optimal diameter prediction model containing the SCI and SDI, and its expression is:

$$D=20.3737SC{I}^{0.0353}{\left[1-\exp (-(0.0908SD{I}^{0.0181})t)\right]}^{2.2214}$$

(18)

3.3.2. Stand Prediction Model of Volume

Similarly,

Table 5. Parameter estimates and fitting statistics of the stand volume models containing the SCI and SDI.

Model No.	Parameterization		Parameter Estimation									Fitting Statistic
	SCI	SDI	a	a1	a2	b	b1	b2	c	c1	c2	RMSE	MARE%	R2
M25	a	b		29.5586	0.7593		0.0615	0.1641	7.3515			34.920	13.63	0.82
M26	a	c		29.4513	0.7618	0.1879				186.8887	−0.4767	38.218	14.91	0.78
M27	b	a		72.1764	0.1386		0.0112	0.9807	2.3595			44.287	17.28	0.69
M28	b	c	203.3057				0.0050	1.1614		26.6712	−0.4231	44.195	17.25	0.69
M29	c	a		69.7764	0.1389	0.1327				207.2330	−1.7921	45.731	17.85	0.66
M30	c	b	197.4059				0.0139	0.2638		71.4353	−1.6122	45.975	17.94	0.66
M31	a	b, c		29.5408	0.7599		0.0841	0.1178		18.9860	−0.1412	38.143	14.89	0.78
M32	b	a, c		117.3496	0.0725		0.0082	1.0495		11.8602	−0.2623	44.039	17.19	0.69
M33	c	a, b		83.5048	0.1138		0.0917	0.0495		185.3114	−1.7620	45.704	17.84	0.66
M34	a, b	c		21.2311	0.8997		0.3367	−0.2472		179.4847	−0.4732	41.374	16.15	0.78
M35	b, c	a		73.6409	0.1364		0.0023	1.6743		0.0444	1.7585	43.694	17.05	0.70
M36	a, c	b		23.3147	0.8599		0.0610	0.1623		1.7588	0.5750	37.807	14.75	0.78

Note: Bold numbers denote the best model for each criterion. a, b and c are the parameters of the base model M5 for volume prediction. For M25, parameters a and b are re-parameterized to $a_{1}SC{I}^{a_2}$ and $b_{1}SD{I}^{b_2}$ by the SCI and SDI, respectively; thus, its equation becomes $V=a_{1}SC{I}^{a_2}[1-\exp (-b_{1}SD{I}^{b_2}t){]}^c$. The other equations can be deduced by analogy.

shows the parameter estimation and fitting statistics of volume models that contain different combinations of the SCI and SDI. The best performing model was M25, which introduced the SCI into parameter a and the SDI into parameter b. M25 had an R² of 0.82, which was 34.80% larger than that of the base model, and the smallest RMSE and MARE, which were 13.409 and 8.38% smaller than the base model, respectively. Therefore, M25 was selected as the optimal volume prediction model at this stage, and its expression is:

$$V=29.5586SC{I}^{0.7593}{\left[1-\exp (-(0.0615SD{I}^{0.1641})t)\right]}^{7.3515}$$

(19)

3.4. Model Construction with Inclusion of Species Composition

3.4.1. Height Prediction Model including the TSCC

Based on the previous optimal height prediction model M6, the parameters were re-parameterized by the TSCC of the three tree species groups; a₁ was transformed into (a₁₁L₁ + a₁₂L₂ + a₁₃L₃), a₂ was transformed into (a₂₁L₁ + a₂₂L₂ + a₂₃L₃), b was transformed into (b₁₁L₁ + b₁₂L₂ + b₁₃L₃), and c was transformed into (c₁₁L₁ + c₁₂L₂ + c₁₃L₃). Then, parameter estimation and fitting criteria calculations were carried out for the height prediction models containing different TSCC combinations, as shown in

Table 6. Fitting statistics of the average height models containing the SCI and TSCC.

Model no.	Parameterization	Fitting Statistic
	TSCC	RMSE	MARE%	R2
M37	a1, a2	1.669	15.05	0.86
M38	a1	1.078	9.72	0.94
M39	b	1.125	10.15	0.94
M40	c	0.798	7.90	0.97
M41	a1, a2, b	1.849	16.68	0.82
M42	a1, b	1.187	10.71	0.93
M43	a1, a2, c	1.032	9.31	0.95
M44	a1, c	0.944	8.52	0.96
M45	b, c	0.974	8.78	0.95
M46	a1, a2, b, c	1.782	16.07	0.84
M47	a1, b, c	1.099	9.91	0.94

Note: Bold numbers denote the best model for each criterion. a₁, a₂, b, and c are parameters of the height prediction model M6. For M37, parameters a₁ and a₂ are re-parameterized to (a₁₁L₁ + a₁₂L₂ + a₁₃L₃) and (a₂₁L₁ + a₂₂L₂ + a₂₃L₃), respectively; thus, its equation becomes $H=(a_{11}{L}_1+a_{12}{L}_2+a_{13}{L}_3)SC{I}^{(a_{21}{L}_1+a_{22}{L}_2+a_{23}{L}_3)}{\left[1-\exp (-bt)\right]}^c$. The other equations can be deduced by analogy.

. Among the models, M40, where L_i was introduced into parameter c, had the largest R² (0.97) and the smallest RMSE (0.798) and MARE (7.90%). Compared with M6, the improvement in the fitting effect of M40 was minimal, with the R² increased by 0.002, RMSE decreased by 0.025, and MARE decreased by 0.03%. Considering the calculation amount and the model convenience, it was not recommended to include the TSCC in the height prediction model.

3.4.2. Diameter Prediction Model including the TSCC

Similarly, parameters a₁, a₂, b₁, b₂, and c of the previous optimal diameter model M13 were re-parameterized by the TSCC of the three tree species groups (L₁, L₂, and L₃).

Table 7. Fitting statistics of the average DBH models containing the SCI, SDI, and TSCC.

Model No.	Parameterization	Fitting Statistic
	TSCC	RMSE	MARE%	R2
M48	a1, a2	1.808	11.77	0.87
M49	a1	1.814	11.80	0.86
M50	b1, b2	1.798	11.70	0.87
M51	b1	1.813	11.80	0.86
M52	c	1.795	11.68	0.87
M53	a1, a2, b1, b2	1.906	12.40	0.85
M54	a1, b1	1.950	12.69	0.84
M55	a1, a2, c	1.552	10.10	0.90
M56	a1, c	1.387	9.02	0.92
M57	b1, b2, c	1.788	11.63	0.87
M58	b1, c	1.767	11.50	0.87
M59	a1, a2, b1, b2, c	1.495	9.73	0.91
M60	a1, b1, c	1.551	10.09	0.90

Note: Bold numbers denote the best model for each criterion. a₁, a₂, b₁, b_2, and c are the parameters of the diameter prediction model M13. For M48, parameters a₁ and a₂ are re-parameterized to (a₁₁L₁ + a₁₂L₂ + a₁₃L₃) and (a₂₁L₁ + a₂₂L₂ + a₂₃L₃), respectively; thus, its equation becomes $D=(a_{11}{L}_1+a_{12}{L}_2+a_{13}{L}_3)SC{I}^{(a_{21}{L}_1+a_{22}{L}_2+a_{23}{L}_3)}{[1-\exp (-b_{1}SD{I}^{b_2}t)]}^c$. The other equations can be deduced by analogy.

shows the fitting criteria for the diameter prediction models containing different TSCC combinations. The model displaying the best statistics was M56, in which L_i was introduced into a₁ and c, with an R² of 0.92, an RMSE of 1.387, and a MARE of 9.02%. Compared with the optimal model M13, into which only the SCI and SDI were introduced, the fitting indices of M56 were improved significantly, with R² increased by 11.07%, and RMSE and MARE decreased by 0.618 and 4.02%, respectively. The expression of M56 is as follows:

$$D=(19.6177L_1+22.3716L_2+18.5414L_3)SC{I}^{0.0521}{\left[1-\exp (-(0.0903SD{I}^{0.0138})t)\right]}^{2.0579L_1+2.7934L_2+1.9120L_3}$$

(20)

3.4.3. Volume Prediction Model including the TSCC

After including the TSCC in the parameters of M25,

Table 8. Fitting statistics of the stand volume models containing the SCI, SDI, and TSCC.

Model No.	Parameterization	Fitting Statistic
	TSCC	RMSE	MARE%	R2
M61	a1, a2	23.645	9.23	0.92
M62	a1	20.317	7.93	0.96
M63	b1, b2	29.620	11.56	0.87
M64	b1	28.535	11.14	0.88
M65	c	22.699	8.86	0.93
M66	a1, a2, b1, b2	26.968	10.52	0.90
M67	a1, b1	30.127	11.76	0.87
M68	a1, a2, c	28.955	11.30	0.88
M69	a1, c	26.445	10.32	0.90
M70	b1, b2, c	28.535	11.14	0.88
M71	b1, c	25.092	9.79	0.91
M72	a1, a2, b1, b2, c	23.988	9.36	0.920
M73	a1, b1, c	21.975	8.58	0.93

Note: Bold numbers denote the best model for each criterion. a₁, a₂, b₁, b_2, and c are the parameters of the volume prediction model M25. For M61, parameters a₁ and a₂ are re-parameterized to (a₁₁L₁ + a₁₂L₂ + a₁₃L₃) and (a₂₁L₁ + a₂₂L₂ + a₂₃L₃), respectively; thus, its equation becomes $V=(a_{11}{L}_1+a_{12}{L}_2+a_{13}{L}_3)SC{I}^{(a_{21}{L}_1+a_{22}{L}_2+a_{23}{L}_3)}[1-\exp (-b_{1}SD{I}^{b_2}t){]}^c$. The other equations can be deduced by analogy.

displays the fitting statistics for the volume prediction models containing different TSCC combinations. The R² of the different models varied, with the largest being M62 (0.96), followed by M73 (0.93), and the smallest being M67 (0.87). Meanwhile, M62 obtained the smallest RMSE and MARE values of 20.317 and 7.93%, respectively. Compared with the optimal volume model M25, into which only the SCI and SDI were introduced, M62 significantly improved all statistical criteria, with R² increased by 17.55%, RMSE and MARE decreased by 14.603 and 5.70%, respectively. The expression of M62 is as follows:

$$V=(28.1049L_1+26.8638L_2+28.9530L_3)SC{I}^{0.7793}\times {\left[1-\exp (-(0.0666SD{I}^{0.1576})t)\right]}^{7.8910}$$

(21)

3.5. Model Applicability Test

By using the validation data, we assessed the applicability of the optimal models in each stage, and the results are displayed in

Table 9. Evaluation statistics of model applicability.

Model No.	Dependent Variable	Evaluation Statistic
		TRS/%	MRE/%	MARE/%	P/%
M5	D	−3.47	3.67	18.44	97.03
M5	H	3.27	3.13	21.73	96.87
M5	V	4.71	4.38	22.74	94.90
M6	H	−0.56	0.60	8.13	99.00
M13	D	1.32	1.12	13.86	97.87
M25	V	1.74	1.74	14.33	96.29
M56	D	−0.58	0.67	9.79	98.41
M62	V	0.73	0.89	8.02	97.82

. The TRS and MRE of all models were within ±5%, the MARE decreased from 18.44%–22.74% to 8.02%–9.79%, and P increased from 94.90%–97.03% to 97.82%–99.00%. The results indicated that all eight models passed the test and were applicable. Among them, M6, M56, and M62 showed the best applicability for predicting H, D, and V, respectively.

Figure 2 shows the comparison of the residual distributions between the base and optimal models for D, H, and V against stand age. The percentages of the residuals of the optimal models for D (M56), H (M6), and V (M62) were reduced significantly more than those of the base models (M5). The residual distribution of each optimal model was relatively uniform, and the absolute value of the residual percentage generally showed a trend of decreasing with the increase in stand age. The residual performance of the optimal models M6, M56, and M62 revealed that the model system developed in this study had good performance and could accurately estimate the growth and yield of Chinese fir mixed forests, especially for mature forests that have reached the age of final felling.

4. Discussion

In the stand height model of Chinese fir mixed forests, the inclusion of the site class index (SCI) improved the fitting effect of the model, confirming the close relationship between stand average height and site quality. According to the biological significance of the Richards equation parameters, parameter a represents the maximum value of tree height growth [31]. We found that adding the SCI to parameter a of the base model had the best fitting effect, which may indicate that the maximum tree height was mainly affected by site quality, and this conclusion has been demonstrated by many studies [6,7,21,32,33]. The addition of the tree species composition coefficient (TSCC) did not significantly improve the accuracy of the height model, possibly because the stand average height of Chinese fir mixed forests was the average height of the dominant tree species, which had little relationship with the species composition but was closely related to the stand site quality [30]. Because adding the TSCC will greatly increase the calculation amount of the model, we do not recommend introducing the TSCC into the average height model of Chinese fir mixed forests. Instead, including only the SCI in the height model would be a better choice.

In the models predicting stand average DBH and volume of Chinese fir mixed forests, the inclusion of the SCI and SDI improved the model performance, and the model fitting statistics were the best when the SCI was introduced into parameter a and the SDI was introduced into parameter b. This indicated that site quality mainly affected the maximum values of stand average DBH and volume, while stand density mainly affected their growth rate. The findings that stand average DBH and volume were affected by site quality and stand density are consistent with the results of other studies [34,35,36,37]. However, these results may also be related to the stand age. In this study, the average stand age was 23 years, which is mature for Chinese fir, and its DBH and volume may be more sensitive to the stand density index [19,24]. Therefore, the relationship between stand yield and the SCI/SDI for young Chinese fir mixed forests remains to be further studied.

Including the tree species composition coefficient (TSCC) in the stand average DBH and volume models significantly improved their accuracy. The best fitting results were obtained when the TSCC was introduced into parameters a₁ and c of the stand average DBH model, and when the TSCC was introduced into parameter a₁ of the stand volume model. The influence of species composition on the average DBH and stand volume were different but had one common aspect, that is, the models containing the TSCC in parameter a₁ of the SCI had the best fitting performance. Therefore, it can be inferred that the composition of tree species will affect the maximum values of the stand average diameter and volume. The reason may be that, because of the greater diversity of tree species, the increased complementary effects among the different tree species will promote the full utilization of space within the stand, thereby increasing the maximum values of the stand average diameter and volume [11,38]. This suggests that the construction of artificial mixed forests can not only improve the ecological benefits but also increase timber production and generate more economic benefits.

In this study, the TSCC was the default fixed value for a certain period of time. However, with the increase in stand age, the interspecific competition within the stand will change, and factors such as growth rate, growth status, and the dominance of different tree species will also lead to changes in the TSCC [16,17,38]. Therefore, follow-up studies can be carried out on the basis of this study, such as developing a dynamic tree species composition coefficient model to provide a more reliable TSCC for predicting the stand dynamics of mixed forests or including air temperature, moisture, nutrients, slope, altitude, management level, and other indicators to build a more accurate stand growth and yield model.

5. Conclusions

In this study, the effects of site quality, stand density, and species composition on the average height, average DBH, and stand volume of mixed Chinese fir forests were explored, and a growth and yield model system was developed. Among them, site quality had a close relationship with the growth limits of stand average height, average DBH, and volume; stand density was closely related to the growth rate of stand average DBH and volume; species composition had an impact on the maximum growth of stand DBH and volume. The accuracy and applicability of the established model system were good, which can provide theoretical support for harvest prediction, tree species matching, and forestry production planning for Chinese fir mixed forests.