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Counting Phylogenetic Networks

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Abstract

We give approximate counting formulae for the numbers of labelled general, treechild, and normal (binary) phylogenetic networks on n vertices. These formulae are of the form \({2^{\gamma n {\rm log}n+O(n)}}\), where the constant \({\gamma}\) is \({\frac{3}{2}}\) for general networks, and \({\frac{5}{4}}\) for tree-child and normal networks. We also show that the number of leaf-labelled tree-child and normal networks with \({\ell}\) leaves are both \({{2}^{2 \ell {\rm log} \ell +O( \ell )}}\). Further we determine the typical numbers of leaves, tree vertices, and reticulation vertices for each of these classes of networks.

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Correspondence to Colin McDiarmid.

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For our friend and colleague James Oxley

The second author was supported by a Canterbury Fellowship at the University of Oxford and the New Zealand Marsden Fund.

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McDiarmid, C., Semple, C. & Welsh, D. Counting Phylogenetic Networks. Ann. Comb. 19, 205–224 (2015). https://doi.org/10.1007/s00026-015-0260-2

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