To resolve a polytomy, Additional file 1: Algorithm S2 first assigns a cluster label to each branch pendant to it, and then builds a tree using DISTIQUE with the cluster labels as leaves; this tree defines a resolution of the polytomy. Given anchor species u,v from two distinct clusters, we compute distances between other pairs of clusters A and B using Eq. (5), defining the quartet frequencies as: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {p}(uv.AB) = \frac {1}{|A||B|}\sum _{a\in A}\sum _{b\in B}{\overline {p}(uv.ab)}.$\end{document}p¯(uv.AB)=1|A||B|∑a∈A∑b∈Bp¯(uv.ab). When all clusters in the consensus tree are correct (expected asymptotically), p(u v.a b) values are identical; thus, all \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {p}(uv.ab)$\end{document}p¯(uv.ab) values are empirical estimates of the same true value, and using their average is justified.