  Recently, an elegant approach in phylogenetics was introduced by
Billera-Holmes-Vogtmann that allows a systematic comparison of different
evolutionary histories using the metric geometry of tree spaces. In many
problem settings one encounters heavily populated phylogenetic trees, where the
large number of leaves encumbers visualization and analysis in the relevant
evolutionary moduli spaces. To address this issue, we introduce tree
dimensionality reduction, a structured approach to reducing large phylogenetic
trees to a distribution of smaller trees. We prove a stability theorem ensuring
that small perturbations of the large trees are taken to small perturbations of
the resulting distributions.
  We then present a series of four biologically motivated applications to the
analysis of genomic data, spanning cancer and infectious disease. The first
quantifies how chemotherapy can disrupt the evolution of common leukemias. The
second examines a link between geometric information and the histologic grade
in relapsed gliomas, where longer relapse branches were specific to high grade
glioma. The third concerns genetic stability of xenograft models of cancer,
where heterogeneity at the single cell level increased with later mouse
passages. The last studies genetic diversity in seasonal influenza A virus. We
apply tree dimensionality reduction to 24 years of longitudinally collected
H3N2 hemagglutinin sequences, generating distributions of smaller trees
spanning between three and five seasons. A negative correlation is observed
between the influenza vaccine effectiveness during a season and the variance of
the distributions produced using preceding seasons' sequence data. We also show
how tree distributions relate to antigenic clusters and choice of influenza
vaccine. Our formalism exposes links between viral genomic data and clinical
observables such as vaccine selection and efficacy.
