Nested tree space: a geometric framework for co-phylogeny

arXiv:2604.05056v2 Announce Type: replace-cross
Abstract: 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 $sigma$-space, a moduli space of fully nested ultrametric phylogenetic trees with a fixed leaf map.
Generalizing the $tau$-space of Gavryushkin and Drummond, $sigma$-space is constructed as a cubical complex parametrised by nested ranked tree topologies and inter-event time coordinates of the combined host and parasite speciation events. We characterise admissible orderings via binary textitnesting sequences and organise them into a natural poset. We show that $sigma$-space is contractible and satisfies Gromov’s cube condition, and is therefore CAT(0). In particular, it admits unique geodesics and well-defined Fr’echet 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.