  Evolutionary relationships between species are represented by phylogenetic
trees, but these relationships are subject to uncertainty due to the random
nature of evolution. A geometry for the space of phylogenetic trees is
necessary in order to properly quantify this uncertainty during the statistical
analysis of collections of possible evolutionary trees inferred from biological
data. Recently, the wald space has been introduced: a length space for trees
which is a certain subset of the manifold of symmetric positive definite
matrices. In this work, the wald space is introduced formally and its topology
and structure is studied in detail. In particular, we show that wald space has
the topology of a disjoint union of open cubes, it is contractible, and by
careful characterization of cube boundaries, we demonstrate that wald space is
a Whitney stratified space of type (A). Imposing the metric induced by the
affine invariant metric on symmetric positive definite matrices, we prove that
wald space is a geodesic Riemann stratified space. A new numerical method is
proposed and investigated for construction of geodesics, computation of
Fr\'echet means and calculation of curvature in wald space. This work is
intended to serve as a mathematical foundation for further geometric and
statistical research on this space.
