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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2601.21578 |
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| _version_ | 1866910004553973760 |
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| author | Liu, Hexuan Lu, Bing-Ze Wu, Taoyang Yu, Guan-Ru |
| author_facet | Liu, Hexuan Lu, Bing-Ze Wu, Taoyang Yu, Guan-Ru |
| contents | This paper studies the enumeration of seven subclasses of level-$2$ phylogenetic networks under various planarity and structural constraints, including terminal planar, tree-child, and galled networks. We derive their exponential generating functions, recurrence relations, and asymptotic formulas. Specifically, we show that the number of networks of size $n$ in each class follows: \[ N_n \sim c \cdot n^{n-1} \cdot γ^n, \] where $c$ is a class-specific constant and $γ$ is the corresponding growth rate. Our results reveal that being terminal planar can significantly reduce the growth rate of general level-2 networks, but has only a minor effect on the growth rates of tree-child and galled level-2 networks. Notably, the growth rate of 3.83 for level-$2$ terminal planar galled tree-child networks is remarkably close to the rate of 2.94 for level-$1$ networks. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_21578 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Asymptotic Enumeration of Subclasses of Level-$2$ Phylogenetic Networks Liu, Hexuan Lu, Bing-Ze Wu, Taoyang Yu, Guan-Ru Combinatorics This paper studies the enumeration of seven subclasses of level-$2$ phylogenetic networks under various planarity and structural constraints, including terminal planar, tree-child, and galled networks. We derive their exponential generating functions, recurrence relations, and asymptotic formulas. Specifically, we show that the number of networks of size $n$ in each class follows: \[ N_n \sim c \cdot n^{n-1} \cdot γ^n, \] where $c$ is a class-specific constant and $γ$ is the corresponding growth rate. Our results reveal that being terminal planar can significantly reduce the growth rate of general level-2 networks, but has only a minor effect on the growth rates of tree-child and galled level-2 networks. Notably, the growth rate of 3.83 for level-$2$ terminal planar galled tree-child networks is remarkably close to the rate of 2.94 for level-$1$ networks. |
| title | Asymptotic Enumeration of Subclasses of Level-$2$ Phylogenetic Networks |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2601.21578 |