Nested tree space: a geometric framework for co-phylogeny
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| Format: | Preprint |
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2026
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| _version_ | 1866913034556932096 |
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| author | Grindstaff, G. Hoekzema, R. S. |
| author_facet | Grindstaff, G. Hoekzema, R. S. |
| contents | 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 Gavryushkin and Drummond, $σ$-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 $σ$-space is contractible and satisfies Gromov's cube condition, and is therefore CAT(0). In particular, it admits unique geodesics and well-defined Fréchet 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_05056 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Nested tree space: a geometric framework for co-phylogeny Grindstaff, G. Hoekzema, R. S. Combinatorics Metric Geometry Populations and Evolution 05C05 92D15 51F99 52B99 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 Gavryushkin and Drummond, $σ$-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 $σ$-space is contractible and satisfies Gromov's cube condition, and is therefore CAT(0). In particular, it admits unique geodesics and well-defined Fréchet 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. |
| title | Nested tree space: a geometric framework for co-phylogeny |
| topic | Combinatorics Metric Geometry Populations and Evolution 05C05 92D15 51F99 52B99 |
| url | https://arxiv.org/abs/2604.05056 |