Compatible taper and volume models were created for
Compatible tapervolume models are flexible tools for estimating total and merchantable tree volume that can meet the demands of market trends as product specifications change. A compatible tapervolume estimation system contains a taper equation and a total volume equation. The taper equation can provide estimations of diameter at a given height up a tree and merchantable tree volume (
Taper and volume estimation systems can be divided into two types: Type (1), the total volume model is directly derived through integration of the taper equation; Type (2), equation form of the total volume model is independent from the taper equation. For type (1), two methods can be used to estimate parameters of the two models: Method (1), firstly fit the taper equation, then the volume model with its parameters can be directly obtained by integration (
A large number of compatible tapervolume systems based on type (1) have been created for oak species in Greece, America, Denmark, Spain and Mexico (
174 trees from 104 plots with an area of 20 × 20 m for cork oak natural forests and plantations in North China were used in this study, 57 of those trees were from plantations (including 31 average trees and 26 dominant trees), and the other 117 trees were from natural forests (including 60 average trees and 57 dominant trees). These plots were created in the following locations with different site conditions and age distributions: GaoLuo forestry station, QiJiahe forestry station, BeiTan forestry station and HengHe forestry station of Zhongtiaoshan region in Shanxi province, collective forests of DaGeliao village in Xingtai city of Hebei province, SiZuolou forestry station and XiShan forestry station in Beijing. Measured and computed variables were as follows: (1) single tree variables including diameter at breast height over bark (
Four alternative modelling strategies were tried, because (1) a few analyzed stems (“trees”) had ramicorns and (2) some of the small analyzed stems had very high values of relative diameter (
For bark thickness models and volume models from the abovementioned four systems, all data were used for model fitting. Before modelling, some explanatory and response variables (
where
To establish bark thickness models and volume models, linear mixed effects analyses were used on transformed variables with the tree number (
where
where
For each model system, an overall meritbased method was used to select model explanatory variables. Regression equations for bark thickness models and volume models with different variable combinations were compared. Four sets of optimal base equations were obtained by examining the coefficients of determination (
For taper modelling of the four systems, a subset of data (80%) from the analyzed stems were randomly selected for the fitting phase, while the remaining data were used for model validation (
where
To make taper modelling simpler, statistic
Similarly, the
To sum up, four sets of data (with a bark thickness model, a volume model and a taper model in each of them) were used for modelling and the most suitable set was then selected using the abovementioned statistics and residual plots.
After the selection of the optimal model system, representing essentially the best dataset, the transformed bark thickness model (
For the taper model in the optimal model system, the predictive performance of
The
Equation forms, coefficients and standard errors of coefficients of the models in System 4 are shown in
Predicted value ranges and residual ranges from the Jackknife validations for the transformed bark thickness model (
Residual plots, predicted value ranges, residual ranges,
Various forms of volume models have been reported in the literature, such as the model represented by
where
In the application of a mixed effect model, when a subsample of the dataset is available to calculate the random effects, users can calibrate the coefficients of the linear mixed effect model (“lme” 
Similar features can be found in the bark thickness model, which was also a linear mixed effect equation using variables transformed by the BoxCox method.
According to several studies in the literature (
A polynomial taper equation (
Branches are an important aspect of tree form because they affect stem utilization. A ramicorn branch is a steepangled branch diverging less than 30° from the main stem and significantly smaller than the main stem (
We built four sets of compatible tapervolume model systems using all the data (Model system 1), using data of stems without ramicorns (Model system 2), data of stems with a
In System 4, data from four big trees were removed because they had ramicorns. Due to the small sample size for big trees, more big trees should be measured in the future to obtain a compatible tapervolume model system with a larger useable diameter span. It should be noted that if models created using System 4 are used for predictions of stems with ramicorn branches, then errors would be likely greater than those reported here. Therefore, we suggest that models created with System 4 can be used for predictions of stems without ramicorn branches and simultaneously with a relative diameter less than 1.5.
Linear mixed effect equations with tree number as random factor were used for bark thickness and volume modelling using variables transformed by the BoxCox method to minimise heteroscedasticity. Using the polynomial equation reported by
Four sets of compatible tapervolume models systems using different data sets were established and compared. The models in System 4 had superior coefficients of determination (
Within the specified ranges of
This research was jointly supported by scientific and research base construction projects of Beijing Municipal Education Commission (SYSBL2009), forestry science promotion project of the State Forestry Bureau (201144), open fund project of Beijing Forestry University “985” advantage subject innovation platform (0001108003), special fund project for forestry public service industry and research (201004021) and China Scholarship Council. We acknowledge the strong support from Zhong Tiaoshan National Forest Authority, Xingtai County Forestry Bureau, Si Zuolou forestry station and Xi Shan forestry station in Beijing.
Conghui Zheng and Yuzhong Wang have equally contributed to this work and should be regarded as cofirst authors.
Loess residual plot of the backtransformed bark thickness model in System 4. The solid horizontal line indicates the baseline, while the red dotted line represents the loess curve.
Loess residual plot of the backtransformed total stem volume inside bark model in System 4. The solid horizontal line indicates the baseline, while the red dotted line represents the loess curve.
Loess residual plot of the backtransformed diameter inside bark model in System 4. The red closed circles represent the residuals under the condition of relative height > 0.9 and the black open circles represent the residuals under the condition of relative height <0.9. The solid horizontal line indicates baseline and the green dotted line indicates loess curve.
Box plot of residuals
Box plot of residuals
Summary statistics of four data sets used for modelling. (
Model type  Modelsystem  Samplenumber  Range ofage (year)  Rangeof 
Range of 
Range of 
Range ofresponsevariable 

1, 3  2358  1684  0.011.50  3.839.9  5.021.0  0.03.5  
2, 4  2059  1684  0.021.50  3.822.6  5.018.2  0.03.0  
1  1299  584  1.00  0.339.9  1.421.0  0.000010.649  
2  1201  584  1.00  0.323.1  1.418.2  0.000010.224  
3  1035  584  1.00  1.639.9  1.421.0  0.00020.649  
4  937  584  1.00  1.623.1  1.418.2  0.00020.224  
1  12814  584  0.0112.00  0.339.9  1.421.0  0.141.0  
2  11336  584  0.0112.00  0.323.1  1.418.2  0.125.5  
3  11419  584  0.011.50  1.639.9  1.421.0  0.141.0  
4  9942  584  0.011.50  1.623.1  1.418.2  0.125.5 
Values of fitting statistics for eight models in four modeling systems. (
Modelstype  Modelsystem  Transformed models 
Backtransformed models 





1, 3  < 2.2e16  0.90  0.22  0.86  0.24  
2, 4  9.646e13  0.94  0.15  0.92  0.15  
1  3.343e12  0.99  0.061  0.99  0.006  
2  9.195e08  0.99  0.043  0.99  0.003  
3  6.921e09  0.99  0.066  0.99  0.006  
4  2.14e06  0.99  0.044  0.99  0.003  
1  < 2.2e16  0.91  0.32  0.97  0.84  
2  < 2.2e16  0.91  0.32  0.97  0.67  
3  < 2.2e16  0.95  0.21  0.97  0.80  
4  < 2.2e16  0.96  0.20  0.98  0.61 
Summaries for the
Model  Equation form  Coefficients (± 
Predicted value range/ Residual range  

Model(Entire data)  Jackknife  


0.881 ± 0.058***  (1.58, 1.22)/(0.61, 0.56)  (1.60, 1.24)/(0.64, 0.60)  

0.277 ± 0.071***  

0.669 ± 0.160***  

0.495 ± 0.101***  

0.278 ± 0.018***  

3.182 ± 0.201***  

8.094 ± 0.882***  

8.695 ± 1.426***  

3.327 ± 0.745***  

0.089 ± 0.022***  
ρ  0.403  
σ^{2}  0.025  
σ_{a0}^{2}  0.013  

    (0.03, 2.56)/(0.60, 0.74)   
Summaries for the
Model  Equation form  Coefficients (± 
Predicted value range/Residual range  

Model  Jackknife  


4.488 ± 0.007***  (4.296,1.244)/(0.218, 0.180)  (4.290,1.233)/(0.243.0.216)  

0.125 ± 0.002***  

0.021 ± 0.005***  
θ  0.467  
σ^{2}  0.002  
σ_{b0}^{2}  0.002  

    (0.0003.0.245)/(0.020, 0.018)   
Summaries for
Model  Equation form  Coefficients (± 
Predicted value range / Residual range  

Entire data  Fit data  Validation data  
Y 

33.972 ± 1.545***  (0.00, 4.04)/(1.18, 1.48)  (0.00, 4.05)/(1.19, 1.24)  (0.00, 3.12)/(1.81, 1.45)  

125.749 ± 8.907***  

229.692 ± 18.913***  

211.380 ± 17.418***  

0.738  

0.359  
σ^{2}  0.043  
σ_{c3}^{2}  0.050; 8.117  
σ_{c4}^{2}  0.210; 24.145  
dib      (0.00, 24.39)/(2.94, 3.03)  (0.00, 24.40)/(2.91, 2.70)  (0.00, 22.94)/(2.60, 3.05) 
Tab. S1  Summary statistics of four data sets used for modelling.
Tab. S2  Values of fitting statistics for eight models (
Tab. S3  Results of
Tab. S4  Results of
Tab. S5  Results of
Tab. S6  The estimated taper function (red dotted curve) and the basic taper data for each system.
Tab. S7  Frequencies,