# Stand Structure Impacts on Forest Modelling
## 5 Height  
Height growth pattern differs by species, site and stand structure [67]. Height is frequently predicted as function of diameter at breast height. A set of functions have been used [2] (pp. 114,115). Due to the aforementioned factors, a wide set of h-d relationships has been developed, mainly for two reasons. First, because diameter is easier to measure more accurately than height [68]. Second, because height is used to predict other variables, such as volume [2]. Height functions should have monotonic increment, inflection point and asymptotical value, which corresponds to sigmoid functions [69]. The different growth patterns of trees by species and by site, as well as by stand structure, derive in a large number from height-diameter functions (h-d) (e.g., [67; 69; 70; 71; 72]). However, most of the functions were developed for monospecific or pure even-aged stands, many of which are monolayers. Thus, their use in multi-layered stands can result in biased estimations of the tree total height in uneven-aged stands [70]. Several authors (e.g., [58]) have demonstrated that some stand variables can improve the h-d function prediction. Considering the tree height and its growth variability, the flexibility of the h-d functions can enable more accurate height predictions (e.g., [70; 73]). Traditionally, h-d functions were fitted using nonlinear least squares regression methods (e.g., [68; 74]). These models are species-, site- and stand structure-specific. More recently, the inclusion of other variables and other regression methods enabled the accommodation of the variability in tree allometry, maintaining model accuracy. Some examples are used to illustrate the influence of the species, stand structure, site and regression methods on h-d functions. [PERSON]. [68] developed h-d functions for 13 riparian species for Missouri, Illinois and Iowa, USA, with nonlinear regression, demonstrating that the equations give reasonable predictions. However, the authors did not give information about the stand structure, indicating that the equations should only be used for the species and the regions where they were fitted. For even-aged stands, Ref. [75] developed h-d models for _Pinus taeda_, for a range of densities using nonlinear regression with one or two independent variables (diameter at breast height, normalised using the Reineke stand density index, SDI). The authors demonstrated that the functions with only diameter at breast height were not able to detect the differences in stem tapper due to density, while the functions with the two independent variables had better predictive abilities. Additionally, for pure even-aged stands, but using mixed models, Ref. [72] stated that for _Tectona grandis_ stands in India, the variability caused by age in h-d with mixed models was low, which was explained by the reduction of competition due to thinning. The effect of thinning intensity, while demonstrating wide variation, was incorporated in the random components of the model. Ref. [76], for _Pinus pinaster_, concluded that the function with mean quadratic diameter and basal area had good performance at stand level and that the predictive ability increased if the trees in the extreme range of distribution were used for the prediction of the random effect. Inversely, Ref. [74], comparing two regression methods (nonlinear regression and generalised additive mixed models) for _Pinus massoriana_ in thinned (from below) and unthinned stands, concluded that the model attained using nonlinear regression outperformed the one fitted with the mixed model. The former was able to accommodate the changes in diameter and height growth of thinning (with thinning from below there was an increase of diameter at breast height growth and a reduction of height growth, the largest the heavier the thinning intensity) due to the reduction of competition. According to Ref. [73], in mixed tropical forests in China, linear and nonlinear regression models had high precision. Ref.