# Verification of the Scots Pine (_Pinus sylvestris_ L.) Crown Length Model
## 3 Results
### Model Verification on Sample Trees  
The allometric formula for calculating crown length (Equation (2)) allows us to estimate the length based on measurable parameters, such as diameter at breast height (\(DBH\)) and tree height (\(H\)). In this equation, the notation: (\(\pi\cdot DGH\cdot H\)) denotes the surface of the cylinder with diameter (\(DGH\)) and height (\(H\)). This equation is particularly useful for trees growing at optimal density \(DI=1\). In such circumstances, a simplified equation is employed, with the compaction index (\(\frac{1}{\sqrt{DI}}\)) being disregarded:  
\[CL_{cal}=(\pi\cdot DGH\cdot H)^{0.25} \tag{3}\]  
The relationships between \(DBH\), \(H\), \(DI\) and \(CL\) for the 300 sample trees are demonstrated in both Table 2 and Figure 4. The results indicated significant correlations for the adopted level of \(p<0.05\), which suggests that the allometric model may be a reliable tool for crown length estimation. For the positive correlation, the highest coefficients of determination occurred between \(CL_{cal}\) and \(DBH\) and \(H\) (\(R^{2}\) = 0.867 and \(R^{2}\) = 0.848, respectively) (Figure 4e,f). In the case of density, a negative linear correlation was observed. The higher the stand density, the shorter the crowns were (Figure 4a,d). The coefficient of determination between \(CL_{emp}\) and \(DI\) was very high (\(R^{2}\) = 0.805). A slightly weaker relationship was noted between \(CL_{cal}\) and \(DI\) (Figure 4d).  
Figure 3: Variability of breast height diameter of pine stands on 20 plots (\(n\) = 10957). On the \(X\)-axis: experimental plot number (1–20); basal area (BA, m\({}^{2}\cdot\)ha\({}^{-1}\)); number of trees in the plot [trees 0.25 ha\({}^{-1}\)] and stand age [years].  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(CL_{emp}\) & \(CL_{cal}\) & \(DBH\) & \(H\) & \(DI\) \\  & **(m)** & **(m)** & **(cm)** & **(m)** & **(-)** \\ \hline \(CL_{cal}\) (m) & 0.8567 & – & & & \\ \(DBH\) (cm) & 0.7503 & 0.9352 & – & & \\ \(H\) (m) & 0.6789 & 0,9227 & 0.7868 & – & \\ \(DI\) (–) & \(-\)0.9273 & \(-\)0.6129 & \(-\)0.4942 & \(-\)0.3975 & – \\ \hline \end{tabular} \(CL_{emp}\), empirical crowns length; \(CI_{cal}\), 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 2: The Spearman rang correlation between empirical crowns length, calculated crowns length, diameter at breast height, height and density index of 300 sample trees. Statistically significant results for \(p<0.05\).  
Figure 4: The relationship between: (**a**) density index and empirical crowns length (**b**) diameter at breast height and empirical crowns length; (**c**) height and empirical crowns length; (**d**) density index and calculated crowns length (**e**) diameter at breast height and calculated crowns length; (**f**) height and calculated crowns length. The red line represent the logarithmic regression (**a,d**) and linear regression (**b,c,e,f**) describing the relationship between the respective explanatory variable and the response. Biometrics for 300 sample trees.