Height–diameter relationship of plantation-grown juvenile black locust trees is differentiated according to their growth rate, which is positively affected by spacing

[i] Abbreviations: d – breast height diameter of the tree (cm), h – total tree height (m). Data range (minimum–maximum) of d and h is shown.

Abbreviations: h – tree height (m), d – diameter at breast height (cm), a₀, a₁, a₂ – model parameters.

* Formulae: mean bias – $\frac{{\displaystyle {\sum }_{i=1}^{\mathit{PM}}\mathit{bias}}}{\mathit{PM}}$ ; Aggregated MSE – mean squared errors aggregated from all plots – ${\displaystyle {\sum }_{i=1}^{\mathit{PM}}({\mathit{bias}}_i^2+{\mathit{var}}_i)}$ ; 90th percentile of the absolute values of the relative biases ARB_i, where bias_i is the mean error per plot ${\mathit{bias}}_i={\displaystyle {\sum }_{j=1}^N\frac{(\widehat{h_j}-h_j)}{N}}$ , var_i is the error variance per plot, ${\mathit{var}}_i={\displaystyle {\sum }_{j=1}^N\frac{{(\widehat{h_j}-h_j)}^2}{N}}$ , ARB_i is the absolute value of the relative bias per plot ${\mathit{ARB}}_i=\left|\frac{{\displaystyle {\sum }_{j=1}^N\frac{(\widehat{h_j}-h_j)}{h_j}}}{N}\right|$ , PM – total number of plots, N – number of the measured tree heights per plot, h_j – observed tree height, $\widehat{h_j}$ – predicted tree height.

[i] Abbreviations: df – degrees of freedom; logLik – log-likelihood.

[i] Abbreviations: ME – model efficiency, $\mathit{ME}=1-\frac{{\displaystyle \sum {(h_i,\widehat{h_i})}^2}}{{\displaystyle \sum {(h_i,\overline{h})}^2}}$ , MARE% is the average of the absolute values of the relative errors ARE%, $\mathit{ARE}\%=\frac{\left|h_i,\widehat{h_i}\right|}{h_i}100$ Bias is the average error, $\mathit{Bias}=\frac{{\displaystyle \sum (h_i,\widehat{h_i})}}{n}$ , where h_i, $\widehat{h_i}$ are experimental and predicted height values of the i-th measurement and h̅ represents the mean observed height value; d – diameter at breast height (cm), h – total tree height (m), Spacing – growing space per plant (m²), a₀, a₁, b₀, b₁ – fixed model parameters, θ, η – variance function parameters, u – random parameter component, ${\sigma }_u^2$ – variance of the random component, σ² – residual variance of the model, SE – standard error.

[i] Abbreviations: u – estimated random component of the rate parameter b₀; Perc0, Perc25, Perc50, Perc75, Perc100 – 0th, 25th, 50th, 75th and 100th percentiles of the relative errors estimated as: $\frac{h_i-\widehat{h_i}}{h_i}$ ; MARE% is the average of the absolute values of the relative errors ARE%, $\mathit{ARE}\%=\frac{\left|h_i-\widehat{h_i}\right|}{h_i}100$ ; Bias is the average error, $\mathit{Bias}=\frac{{\displaystyle \sum (h_i-\widehat{h_i})}}{n}$ , where h_i, $\widehat{h_i}$ are experimental and predicted height values of the i-th measurement and h̅ represents the mean observed height value.