  The phylogenetic tree space introduced by Billera, Holmes, and Vogtmann (BHV
tree space) is a CAT(0) continuous space that represents trees with edge
weights with an intrinsic geodesic distance measure. The geodesic distance
measure unique to BHV tree space is well known to be computable in polynomial
time, which makes it a potentially powerful tool for optimization problems in
phylogenetics and phylogenomics. Specifically, there is significant interest in
comparing and combining phylogenetic trees. For example, BHV tree space has
been shown to be potentially useful in tree summary and consensus methods,
which require combining trees with different number of leaves. Yet an open
problem is to transition between BHV tree spaces of different maximal
dimension, where each maximal dimension corresponds to the complete set of
edge-weighted trees with a fixed number of leaves. We show a combinatorial
method to transition between copies of BHV tree spaces in which trees with
different numbers of taxa can be studied, derived from its topological
structure and geometric properties. This method removes obstacles for embedding
problems such as supertree and consensus methods in the BHV treespace
framework.
