To test for the additivity of a distance matrix D, we can use the four point condition . For a quartet of leaves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$Q=\{a,b,c,d\}\subset \mathcal {L}$\end{document}Q={a,b,c,d}⊂ℒ, the median and the maximum of the following three values should be the same: {D[a,b]+D[c,d],D[a,c]+D[b,d],D[a,d]+D[b,c]}. When internal branch lengths are assumed positive, as we do throughout this paper, the minimum value is strictly smaller than the median. Assuming w.l.o.g. D[a,b]+D[c,d] is the smallest value, we can infer a b.c d is the topology induced by T D. Let τ(Q)>0 denote the length of the single internal branch in this quartet tree, which we call its “quartet length”; i.e., if \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$ab.cd \in \mathcal {Q}^{T}$\end{document}ab.cd∈QT, then \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\tau (Q)= \frac {1}{2} (D[a,c] + D[b,d] - D[a,b] - D[c,d])$\end{document}τ(Q)=12(D[a,c]+D[b,d]−D[a,b]−D[c,d]).