Towering BHV spaces to analyse trees with non-identical leaves

Music City Center 

Phylogenetic trees represent the shared evolutionary history of organisms, and collections of these trees arise in modern biodiversity studies. The Billera-Holmes-Vogtmann (BHV) space is a non-Euclidean continuous metric space whose metric compares trees with respect to both their discrete structure and branch lengths. Remarkably, while the number of possible topologies grows super-exponentially in the number of leaves, the geometrical properties of BHV space admit a polynomial-time algorithm for distance computations. However, BHV distances are only defined between trees with identical leaf sets, limiting their use in practice. To address this, we propose an extension of BHV space to compare trees with overlapping, but not identical, leaf sets. Our proposal consists of "towers" of BHV spaces, where tower levels are traversed by adding or removing leaves with external branches of length zero adjacent to internal edges. We discuss the construction of the "Towering Tree Space", its geometric properties, distance computation, and statistical analysis using the towering metric. This is joint work with Amy Willis.