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Rooted and ranked binary trees are mathematical objects of great importanceused to model hierarchical data and evolutionary relationships withapplications in many fields including evolutionary biology and geneticepidemiology. Bayesian phylogenetic inference usually explore the posteriordistribution of trees via Markov Chain Monte Carlo methods, however assessinguncertainty and summarizing distributions or samples of such trees remainschallenging. While labelled phylogenetic trees have been extensively studied,relatively less literature exists for unlabelled trees which are increasinglyuseful, for example when one seeks to summarize samples of trees obtained withdifferent methods, or from different samples and environments, and wishes toassess stability and generalizability of these summaries. In our paper, weexploit recently proposed distance metrics of unlabelled ranked binary treesand unlabelled ranked genealogies (equipped with branch lengths) to define theFrechet mean and variance as summaries of these tree distributions. We providean efficient combinatorial optimization algorithm for computing the Frechetmean from a sample of or distribution on unlabelled ranked tree shapes andunlabelled ranked genealogies. We show the applicability of our summarystatistics for studying popular tree distributions and for comparing theSARS-CoV-2 evolutionary trees across different locations during the COVID-19epidemic in 2020.

Statistical summaries of unlabelled evolutionary trees and ranked hierarchical clustering trees