  The metric space of phylogenetic trees defined by Billera, Holmes, and
Vogtmann, which we refer to as BHV space, provides a natural geometric setting
for describing collections of trees on the same set of taxa. However, it is
sometimes necessary to analyze collections of trees on non-identical taxa sets
(i.e., with different numbers of leaves), and in this context it is not evident
how to apply BHV space. Davidson et al. recently approached this problem by
describing a combinatorial algorithm extending tree topologies to regions in
higher dimensional tree spaces, so that one can quickly compute which
topologies contain a given tree as partial data. In this paper, we refine and
adapt their algorithm to work for metric trees to give a full characterization
of the subspace of extensions of a subtree. We describe how to apply our
algorithm to define and search a space of possible supertrees and, for a
collection of tree fragments with different leaf sets, to measure their
compatibility.
