  A phylogenetic tree is an acyclic graph with distinctly labeled leaves, whose
internal edges have a positive weight. Given a set of n leaves, the collection
of all phylogenetic trees with this leaf set can be assembled into a metric
cube complex known as phylogenetic tree space, or Billera-Holmes-Vogtmann tree
space. In this largely combinatorial paper, we show that the isometry group of
this space is the symmetric group on n elements. This fact is relevant to
distance-based analyses of phylogenetic tree sets.
