Table 9 PBIRROL model—sub-models for tree variables prediction (tree height, height index, tree age and average tree crown ratio)

Tree height equation

 \( h={h_{\mathrm{dom}}}\,\left( {1+a{e^{{0.0833{h_{\mathrm{dom}}}}}}} \right)\,\left( {1-{e^{{-1.0959\frac{d}{{{h_{\mathrm{dom}}}}}}}}} \right) \)

 with \( \alpha =0.0509+0.0528\frac{N}{1,000 }+0.00488{dg}-0.00553{d_{\mathrm{dom}}}+0.00036G>d \)

 R ² = 0.872; \( R_{\mathrm{adj}}^2 \) = 0.872; RMS = 1.824; mean PRESS = −0.009; mean APRESS = 1.045; n = 4,215

Height index equation for site evaluation

 \( {s_h}25=1.3+\left( {h-1.3} \right)\frac{{\left( {1-{e^{-1.1725 }}} \right)}}{{\left( {1-{e^{-0.0469d }}} \right)}} \)

 Guide curve as \( h=1.3+20.3423\left( {1-{e^{-0.0469d }}} \right) \)

 R ² = 0.759; \( R_{\mathrm{adj}}^2 \) = 0.759; RMS = 3.563; mean PRESS = −0.019; mean APRESS = 1.477; n = 2,783

Tree age equation

 \( t=\frac{1}{-0.0147}\ln \left[ {\frac{{\frac{{-6.893\mathrm{E}8}}{{592.8+8.9809d+15.6398{dg}-10.7488{d_{\mathrm{dom}}}+2.017G>d-377\frac{h}{{{h_{\mathrm{dom}}}}}+372\frac{h}{{{S_h}25}}}}-1}}{-1,545,918 }} \right] \)

 R ² = 0.730; \( R_{\mathrm{adj}}^2 \) = 0.728; RMS = 26.918; mean PRESS = −0.001; mean APRESS = 4.136; n = 880

Average tree crown ratio equation

 \( \bar{{c}} {r}=1-{e^{{-{{{\left( {-1.1414+0.000629{h_{\mathrm{dom}}}+0.000048N-0.00148G+0.00933\bar{h}+0.00164\bar{t}} \right)}}^{10 }}}}} \)

 R ² = 0.706; \( R_{\mathrm{adj}}^2 \) = 0.688; RMS = 0.003; mean PRESS = 0.0004; mean APRESS = 0.042; n = 90