Tree height and diameter at breast height (DBH, 1.3 m above ground) are fundamental essential individual tree variables used in forest measurements and inventories (Adamec and Drápela, 2015; Misir, 2010; Subedi and Sharma, 2011). DBH can be easily and accurately measured in field inventories, whereas due to visual obstructions the estimation of tree height is less accurate and often far more difficult and time consuming, especially in tall and closed-canopy stands (Ahmadi et al., 2013; Bi et al., 2012). Therefore, height–diameter (H-D) models are commonly used to predict the heights of the remaining trees, thus reducing the cost of data acquisition (Crecente-Campo et al., 2010; Schmidt et al., 2011).

The development of a suitable H-D model is very important for estimating tree growth, biomass production, and carbon storage capacity, as well as for forest management and monitoring (Aishan et al., 2016; Schmidt et al., 2011; Vospernik et al., 2010). A wide variety of models have been proposed to determine the H-D relationships in different tree species, regions, and ecological zones (Feldpausch et al., 2011; Temesgen et al., 2014). Generally, the models can be divided into linear and nonlinear types, but the H-D relationship is influenced by numerous factors, particularly site quality and silvicultural treatments. In many cases, such relationships feature a complex nonlinear process, which is difficult to describe using ordinary linear models (Adamec and Drápela, 2015; Nanos et al., 2004; Watt and Kirschbaum, 2011). Thus, there is a general consensus that nonlinear models can improve measurement accuracy and reliability, leading to better predictions (Cimini and Salvati, 2011; Crecente-Campo et al., 2010; Paulo et al., 2011). Because many H-D models have been proposed in the literature, comparisons among different models are required to identify appropriate H-D relationships.

Cryptomeria fortunei Hooibrenk ex Otto et Dietr is a widespread and economically important coniferous species in plantations in Guizhou Province, China. Because of its relatively fast growth and good-quality timber, as well as its ability to grow in harsh conditions, it plays an important role in timber production, carbon storage, and ecological restoration. Traditionally simple, low-efficiency, and high-consumption management modes have been operated for a long time; however, this has seriously restricted the development and extension of C. fortunei plantations. For these reasons, there is a need to monitor and accurately estimate the growth and yield, and develop a rational extension plan for commodity production and the provision of ecological services. To achieve this, it is necessary to explore the H-D relationship for estimating tree growth and biomass production. No generalized H-D model for C. fortunei has yet been developed or proposed in the literature.

In this study, 16 nonlinear growth functions were fitted to H-D data for a C. fortunei plantation in the Pingba region of Guizhou Province. Four statistical criteria were applied to assess the model performance, and 20 % of an independent dataset was used to validate and compare the performance of each model. The objectives of this study were to (1) compare and evaluate the relative performance of 16 classical nonlinear H-D models and (2) develop appropriate H-D models for a C. fortunei plantation in the Pingba region of Guizhou Province.

Most of the candidate models (Table 2) performed quite well in fitting the H-D relationship for a C. fortunei plantation based on an analysis of the multimodel performance criteria. When selecting a functional form for the H-D relationship, the following features should be considered: prediction effect and precision, and the function's mathematical properties and its biological interpretation (Huang et al., 1992; Misir, 2010; Stankova and Diéguez-Aranda, 2012). Although any function may be considered superior or inferior in a particular situation, in general, we found that the Chapman–Richards, Weibull, and Näslund models proved more effective for predicting the tree height of C. fortunei than other candidate models, because they had a relatively low RMSE, RMA, and AIC, and high Radj2 values. To a certain extent, our results are consistent with findings reported previously (Crecente-Campo et al., 2014; Jiang and Li, 2010; Rupšys, 2015; Scaranello et al., 2012). Huang et al. (1992) fitted and evaluated twenty nonlinear H-D functions for major Alberta species based on a dataset consisting of 13 489 trees of 16 different species, and they found that the Chapman–Richards and Weibull functions gave the most satisfactory results. Mehtätalo et al. (2015) reported that an extensive comparison of 16 nonlinear functions using a total of 28 datasets from a wide range of tree species, regions, and ecological zones confirmed that the Näslund function provided a satisfactory fit in most datasets for the H-D relationship. We think the reason for this is that the recommended functions also have the flexibility to assume various shapes with different parameter values and produce satisfactory curves under most circumstances, especially with their extensive application. In addition, the relationship between tree height and DBH could be described by an S-shape curve, where the tree height generally grew with the increase of DBH and then stabilized. The above three functions are sigmoidal models, with an asymptote that can characterize this kind of phenomenon (Huang et al., 1992). Therefore, we recommend that the Chapman–Richards, Weibull, and Näslund models be applied for accurate predictions of the H-D relationship for C. fortunei plantations in the Pingba region of Guizhou Province.

The differences between the Chapman–Richards, Weibull, and Näslund models are that the Chapman–Richards and Weibull functions are three-parameter functions, while the Näslund function is a two-parameter function. It was found that three-parameter functions are often most appropriate for large datasets; however, the three-parameter models sometimes suffer from problems regarding model convergence and a non-positive definite random-effect variance–covariance matrix (Mehtätalo et al., 2015), while our results also showed that three-parameter models always had high levels of accuracy. Nevertheless, two-parameter models are easy to fit and are quick to achieve convergence in most situations. Therefore, we suggest that the Chapman–Richards and Weibull three-parameter models should be applied when there are large amounts of data to handle and high levels of accuracy are required, while the Näslund two-parameter model should be used when there is limited data available and a relatively lower level of accuracy can be accepted. This is because the Näslund computing processes are often more simple, stable, and efficient than those of the Chapman–Richards and Weibull models.

Many years of forestry practice have confirmed that C. fortunei can be cultivated in central Guizhou Province on a large scale, and this will continue due to its good growth performance and excellent development potential. These proposed generalized H-D models will help forest managers to reduce costs and save time during the inventory process and at the same time to make better decisions in forest management and planning. A generalized H-D model, with a wider range of geographical applicability, can be built by considering stand variables (Ahmadi et al., 2013; Crecente-Campo et al., 2010). Although we proposed three generalized H-D models for use with a dataset that encompassed a wide range of stand ages, there was a clear limitation concerning the number of stands used in the analysis. This was because the C. fortunei plantation in our study region was mostly distributed at middle-altitude sites, with a gentle slope, and the differences among habitats were relatively small. The incorporation of variables that describe site-specific conditions and how such models might relate to the relatively more empirical mixed-model approach implemented here should be investigated in the future.