  Inferring evolutionary histories (phylogenetic trees) has important
applications in biology, criminology and public health. However, phylogenetic
trees are complex mathematical objects that reside in a non-Euclidean space,
which complicates their analysis. While our mathematical, algorithmic, and
probabilistic understanding of phylogenies in their metric space is mature,
rigorous inferential infrastructure is as yet undeveloped. In this manuscript
we unify recent computational and probabilistic advances to construct
tree--valued confidence sets. The procedure accounts for both centre and
multiple directions of tree--valued variability. We draw on block replicates to
improve testing, identifying the best supported most recent ancestor of the
Zika virus, and formally testing the hypothesis that a Floridian dentist with
AIDS infected two of his patients with HIV. The method illustrates connections
between variability in Euclidean and tree space, opening phylogenetic tree
analysis to techniques available in the multivariate Euclidean setting.
