  Rooted and ranked binary trees are mathematical objects of great importance
used to model hierarchical data and evolutionary relationships with
applications in many fields including evolutionary biology and genetic
epidemiology. Bayesian phylogenetic inference usually explore the posterior
distribution of trees via Markov Chain Monte Carlo methods, however assessing
uncertainty and summarizing distributions or samples of such trees remains
challenging. While labelled phylogenetic trees have been extensively studied,
relatively less literature exists for unlabelled trees which are increasingly
useful, for example when one seeks to summarize samples of trees obtained with
different methods, or from different samples and environments, and wishes to
assess stability and generalizability of these summaries. In our paper, we
exploit recently proposed distance metrics of unlabelled ranked binary trees
and unlabelled ranked genealogies (equipped with branch lengths) to define the
Frechet mean and variance as summaries of these tree distributions. We provide
an efficient combinatorial optimization algorithm for computing the Frechet
mean from a sample of or distribution on unlabelled ranked tree shapes and
unlabelled ranked genealogies. We show the applicability of our summary
statistics for studying popular tree distributions and for comparing the
SARS-CoV-2 evolutionary trees across different locations during the COVID-19
epidemic in 2020.
