Given two positive constants α,βand a monotonically increasing function f(x) bounded above by β for positive x (i.e., 0<f(x)<β for x>0), two “anchor” leaves \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$u,v\in \mathcal {L}$\end{document}u,v∈ℒ, and a tree T equivalent to distance matrix D with the corresponding quartet length function τ(Q), we define: 1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\begin{array}{@{}rcl@{}} D^{\prime}_{uv} [a,b] &=& \left\{ \begin{array}{ll} \beta + \alpha. \tau(\{a,b,u,v\}) & ab.uv \notin \mathcal{Q}^{T} \\ \beta - f(\tau(\{a,b,u,v\})) & ab.uv \in \mathcal{Q}^{T} \end{array}\right. \end{array} $$ \end{document}Duv′[a,b]=β+α.τ({a,b,u,v})ab.uv∉QTβ−f(τ({a,b,u,v}))ab.uv∈QT