# Verification of the Scots Pine (_Pinus sylvestris_ L.) Crown Length Model
## 4 Discussion  
The difficulty in modeling crown sizes is the result of cumulative events and conditions that occurred in previous years [51]. Pine crown length is shaped, among others, by intermediate cutting over many years of the tree's life cycle. As a result, certain tree parts affecting density and light conditions inside the stands are removed [21; 25], which is what mainly determines crown shapes and sizes. The size of the crown significantly correlates with other tree parameters, which is why crown measures are important predictors in the development of tree and biomass growth models [17]. Checking the equation is a crucial modeling procedure as it ensures the model's reliability and confidence. When the construction of the model and its verification is carried \begin{table}
\begin{tabular}{c c c c c c} \hline  & \(CL_{emp}\) & \(CL_{cal}\) & \(DBH\) & \(H\) & \(DI\) \\  & **(m)** & **(m)** & **(cm)** & **(m)** & **(–)** \\ \hline \(CL_{cal}\) (m) & 0.9549 & – & & & \\ \(DBH\) (cm) & 0.9038 & 0.9549 & – & & \\ \(H\) (m) & 0.8947 & 0.9669 & 0.9128 & – & \\ \(DI\) (\(-\)) & \(-\)0.8874 & \(-\)0.7390 & \(-\)0.6719 & \(-\)0.6350 & – \\ \hline \end{tabular} \(CL_{emp}\), empirical crowns length; \(CL_{cal}\), calculated crowns length; \(DBH\), diameter at breast height (1.3 m above ground level); \(H\), trees height; \(DI\), density index (actual number of trees per ha/normative number of trees (taken from Tables of Stands Volume and Increment [48]).  
\end{table} 
Table 4: The Spearman rank correlation between averaged values of biometric parameters (\(CL_{emp}\), \(CL_{cal}\), \(DBH\), \(H\), \(DI\)) on 20 plots. Statistically significant results for \(p\) < 0.05. Figure 6: The relationship between: (**a**) empirical crowns length (\(CL_{emp}\)) and calculated (\(CL_{cal}\)) (for 300 _Pinus sylvestris_ L. model trees); (**b**) empirical crowns length (\(CL_{emp}\)) a calculated (\(CL_{cal}\)) (for 20 stands of _Pinus sylvestris_ L). The red line represent the linear regression describing the relationship between the respective explanatory variable and the response. out on the same object, it can be expected that they would significantly correlate with each other, and the assumption of independent errors is compromised when estimating a model with a simple least square regression [52]. This results in a significantly biased parameter estimates that cause invalidation of the hypothesis tests [16; 53]. A correct solution to this problem is to verify the model on diverse data coming from various independent objects, not related to those used for its construction. This way, verification takes into account the heterogeneity and randomness caused by various stochastic factors [16; 52; 54]. The tested \(CL_{cal}\) model can work well in overpopulated and thinned communities. The analysis shows that \"the square of the crown length is directly proportional to the geometric mean height and diameter of the tree at the ground level\". Such a relationship is a representation of biological processes occurring for many years in the phytocenose of pine. Themodel [14] requires \(DBH\) and \(H\) measurements, as well as the determination of \(DI\) as input variables. We tested it on a different dataset than the one used in its construction. Empirical data for assessing the accuracy of the model were collected from different areas, which resulted in their diversification even when the age of the stands was the same (Figure 2), which was confirmed by the ANOVA test. This variability was mainly due to the care procedures performed, as well as habitat conditions and biotic factors. Predictive statistics such as \(R^{2}=0.76\) and RMSE = 31 cm, calculated for the averaged crown lengths, as well as graphical matching curves, showed that the model worked within the assumed error limits. The analysis showed greater agreement between \(CL_{cal}\) and \(CL_{emp}\) for a stand than for a single tree (Figure 6), thereby supporting our hypothesis.