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arXiv · 2604.05056

Nested tree space: a geometric framework for co-phylogeny

G. Grindstaff·R. S. Hoekzema
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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 \textit{nesting 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.

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BibTeXRIS

G. Grindstaff, R. S. Hoekzema. 2026-04-06. Nested tree space: a geometric framework for co-phylogeny. https://arxiv.org/abs/2604.05056

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