Annals of Combinatorics

, Volume 19, Issue 1, pp 205–224 | Cite as

Counting Phylogenetic Networks

  • Colin McDiarmidEmail author
  • Charles Semple
  • Dominic Welsh


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.

Mathematics Subject Classification

05C30 92D15 05C80 


phylogenetic networks tree-child networks normal networks 


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Copyright information

© Springer Basel 2015

Authors and Affiliations

  • Colin McDiarmid
    • 1
    Email author
  • Charles Semple
    • 2
  • Dominic Welsh
    • 3
  1. 1.Department of StatisticsUniversity of OxfordOxfordUK
  2. 2.School of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand
  3. 3.Mathematical InstituteUniversity of OxfordOxfordUK

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