Representations of Partial Leaf Sets in Phylogenetic Tree Space

The metric space of phylogenetic trees defined by Billera, Holmes, and Vogtmann [ Adv. in Appl. Math., 27 (2001), pp. 733--767], 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 nonidentical taxa sets (i.e., with different numbers of leaves), and in this context it is not evident how to apply BHV space. Ren et al. [ A Combinatorial Method for Connecting BHV Spaces Representing Different Numbers of Taxa, preprint, 2017] approach 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.