Smoothing For species tree branches that are even moderately long, expected frequencies of alternative quartet topologies become exceedingly close to zero. For example, a species tree quartet length of 12 in coalescent units  results in a 99.6 % chance of observing no discordance among 1000 genes. Thus, our simple empirical frequency estimator \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\overline {p}$\end{document}p¯ can easily be equal to zero, resulting in distances of infinity (Eq. 5). To avoid this problem, we use Krichevsky-Trofimov  (i.e., add-half estimator), which adds a pseudo-count of 0.5 for each of three possible quartet topologies. This estimator has been shown to reach the min-max cumulative loss for KL divergence asymptotically .