Table 2 Taper equations evaluated for predicting upper stem diameter in this study

M1: Cervera 1973

\( \frac{d}{D}=\left({b}_1+{b}_2X+{b}_3{X}^2+{b}_4{X}^3+{b}_5{X}^4\right) \)

M2: Kozak 1988

\( d={b}_1{D}^{b_2}{b_3}^D{\left(\frac{1-\sqrt{t}}{1-\sqrt{p}}\right)}^{\left({b}_4{t}^2+{b}_5\ln \left(t+0.001\right)+{b}_6\sqrt{t}+{b}_7\exp (t)+{b}_8\left(\frac{D}{H}\right)\right)} \)

M3: Bi 2000

\( \frac{d}{D}={\left(\frac{\ln \left(\sin \left(\frac{\uppi}{2}t\right)\right)}{\ln \left(\sin \left(\frac{\uppi}{2}\frac{1.37}{H}\right)\right)}\right)}^{b_1+{b}_2\sin \left(\frac{\uppi}{2}t\right)+{b}_3\cos \left(\left(\frac{3\uppi}{2}\right)t\right)+{b}_4\left(\frac{\sin \left(\left(\frac{\uppi}{2}\right)t\right)}{t}\right)+{b}_5D+{b}_6t\sqrt{D}+{b}_7t\sqrt{H}} \)

M4: Kozak 2004

\( d={b}_1{D}^{b_2}{H}^{b_3}{\left[\frac{1-{t}^{\frac{1}{3}}}{1-{k}^{\frac{1}{3}}}\right]}^{b_4{t}^4+{b}_5\left(\frac{1}{\mathit{\exp}\left(\frac{D}{H}\right)}\right)+{b}_6{\left(\frac{1-{t}^{\frac{1}{3}}}{1-{k}^{\frac{1}{3}}}\right)}^{0.1}+{b}_7\left(\frac{1}{D}\right)+{b}_8\left({H}^{1-{\left(\frac{h}{H}\right)}^{\frac{1}{3}}}\right)+{b}_9\left(\frac{1-{t}^{\frac{1}{3}}}{1-{k}^{\frac{1}{3}}}\right)} \)

M5: Arias-Rodil et al. 2014

\( d=2\left\{\frac{b_1D}{1-\exp \left({b}_3\left(1.3-H\right)\right)}+\left(\frac{D}{2}-{b}_1D\right)\times \left[1-\left(\frac{1}{1-\exp \left({b}_2\left(1.3-H\right)\right)}\right)\right]+\exp \left(-{b}_2h\right)\times \left[\frac{\left(\frac{D}{2}-{b}_1D\right)\exp \left(1.3{b}_2\right)}{1-\exp \left({b}_2\left(1.3-H\right)\right)}\right]-\exp \left({b}_3h\right)\left[\frac{\left({b}_1D\ \exp \left(-{b}_3H\right)\right)}{1-\exp \left({b}_3\left(1.3-H\right)\right)}\right]\right\} \)