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 $$d=2\left\{\frac{b_1D}{1-\exp \left({b}_3\left(1.3-H\right)\right)}+\left(\frac{D}{2}-{b}_1D\right)\times \left[1-\left(\frac{1}{1-\exp \left({b}_2\left(1.3-H\right)\right)}\right)\right]+\exp \left(-{b}_2h\right)\times \left[\frac{\left(\frac{D}{2}-{b}_1D\right)\exp \left(1.3{b}_2\right)}{1-\exp \left({b}_2\left(1.3-H\right)\right)}\right]-\exp \left({b}_3h\right)\left[\frac{\left({b}_1D\ \exp \left(-{b}_3H\right)\right)}{1-\exp \left({b}_3\left(1.3-H\right)\right)}\right]\right\}$$
1. where, D is the diameter at breast height outside bark (cm); h is the height above ground level (m); d is the diameter inside bark (cm) at height h; H is the total tree height (m); b1b9 are regression coefficients to be estimated from data; $$X=\frac{H-h}{H-1.3}$$; t = h/H, k = 1.3/H; p is the inflection point and was set to 0.25 and 0.12 for Douglas-fir and western hemlock, respectively