In Table 2, for each of the seven cases, we compute L,R1,R2. We use Eq. (1) to derive \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[x,y]$\end{document}Duv′[x,y] values. Where x y.u v is induced by the tree shown in Fig. 4, we use [β−f(t)] and otherwise we use [β+α t], where t=τ(x,y,u,v) is the length of the internal branch for the quartet tree induced by {x,y,u,v}. For example, for Case 1, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b]=[\beta -f(e_{1}+e_{3})]$\end{document}Duv′[a,b]=[β−f(e1+e3)] because a b.u v is induced by the tree, and the length of the edge on the a b.u v quartet tree is e 1+e 3; in Case 7, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$D^{\prime }_{uv}[a,b]=\beta +\alpha e_{1}$\end{document}Duv′[a,b]=β+αe1 because a b.u v is not induced by the tree and τ(a,b,u,v)=e 1.