Nested Tree Space: a geometric framework for co-phylogeny
Gillian Grindstaff, Renee S. Hoekzema
Nested (or reconciled) phylogenetic trees model co-evolutionary systems in which one evolutionary history is embedded within another. We introduce a geometric framework for such systems by defining σ-space, a moduli space of fully nested ultrametric phylogenetic trees with a fixed leaf map. Generalizing the τ -space of Gavyushkin and Drummond, σ-space is constructed as a cubical complex parametrised by nested ranked tree topologies and interevent time coordinates of the combined host and parasite speciation events. We characterise admissible orderings via binary nesting sequences and organise them into a natural poset. We show that σ-space is contractible and satisfies Gromov’s cube condition, and is therefore CAT(0). In particular, it admits unique geodesics and well-defined Frechet means. We further describe its geometric structure, including boundary strata corresponding to cospeciation events, and relate it to products of ultrametric tree spaces via natural forgetful maps.