# Verification of the Scots Pine (_Pinus sylvestris_ L.) Crown Length Model
## 3 Results
### Testing the Crowns Length Model on Averaged Samples  
The effectiveness of the \(CL_{cal}\) model was analyzed on the data averaged for a given stand. Table 3 presents the actual mean \(H\), \(DBH\), and \(CL_{emp}\) parameters of the sample trees, which were correlated with the density of pine stands. The model was employed to calculate the average crowns lengths for 20 analyzed stands. Root mean squared error (RMSE) was used as a measure of the difference between the measured and calculated crowns length values. The difference between the actual and model results was 31 cm, which constituted a discrepancy of 4.57%. This difference was influenced by parameters of the trees collected from three plots: 8, 11, and 16 (Table 3). The mean empirical crowns length was 5.12 m, while the mean crowns length calculated using the verified model was 5.06 m (Table 3). The paired \(t\)-test also showed no significant difference between the mean values of the measured and model-calculated crown lengths for the 20 studied pine stands (t = 0.76; df = 19; \(p\) = 0.45). The analysis of multivariate data for the mean values of 300 sample trees is shown in Figure 5.  
Results for the actual and model-assessed crowns lengths demonstrated that \(R^{2}\) values ranged from 0.519 to 0.972 (\(p<0.001\)). High values of correlation coefficients between various parameters (\(CL\), \(DBH\), \(H\), \(DI\)) of the stands indicated strong relationships between the features that shape tree growth dynamics. This was also confirmed by [PERSON]'s rank correlation analysis (Table 4). A very strong correlation was found between crown length and diameter at breast height and tree height. For \(CL_{emp}\), this relationship was \(R\) = 0.904 and \(R\) = 0.895, respectively. Even stronger correlations of these relationships were found for \(CL_{cal}\), where they were \(R\) = 0.955 and \(R\) = 0.967 (Table 4). The relationship between crown length and stand density is of particular interest. When comparing the correlation  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Number of Plots** & **Mean** & **Mean** & **DI** & **Mean** & **Mean** & **RMSE** & **Divergence** & \(\mathbf{R^{2}}\) & \(\mathbf{N}\) \\  & **\(\mathbf{DGH}\)** & **\(\mathbf{H}\)** & **(\(\mathbf{c}\))** & **\(\mathbf{CL_{emp}}\)** & **(\(\mathbf{c}\))** & **(\(\mathbf{m}\))** & **(\(\mathbf{m}\))** & **(\%)** & **(\(\mathbf{  
for individual 300 sample trees (Table 2) with the correlation for results averaged for 20 separate stands, we observed stronger correlations between the density index and crown length at the population level (\(R=-0.887\) for \(CL_{emp}\) and \(R=-0.739\) for \(CL_{cal}\)) (Table 4). This finding is consistent with the assumption that in single-species stands, such as those of _Pinus sylvestris_, crown length depends mainly on tree density. Figure 5: The relationship between: (**a**) density index and mean empirical crowns length (**b**) mean diameter at breast height and mean empirical crowns length; (**c**) mean height and mean empirical crowns length; (**d**) density index and mean calculated crowns length (**e**) mean diameter at breast height and mean calculated crown length; (**f**) mean height and mean calculated crowns length. The red line represent the linear regression describing the relationship between the respective explanatory variable and the response. The linear relationship showed a very strong correlation between the actual and model-calculated crown lengths at the population level, with a coefficient of \(R\) = 0.949. A comparison of the dispersion of the empirical values (\(CL_{emp}\)) with the calculated ones (\(CL_{cal}\)) for the stands and model trees is presented in Figure 6. High efficiency of the model was confirmed by the coefficient of determination (\(R^{2}\) = 0.901) for the studied tree population (Figure 6b), which proved the validity of the model construction and the accuracy of parameter selection.