Journal of Mathematical Biology

, Volume 78, Issue 4, pp 899–918 | Cite as

Tree-based networks: characterisations, metrics, and support trees

  • Joan Carles PonsEmail author
  • Charles Semple
  • Mike Steel


Phylogenetic networks generalise phylogenetic trees and allow for the accurate representation of the evolutionary history of a set of present-day species whose past includes reticulate events such as hybridisation and lateral gene transfer. One way to obtain such a network is by starting with a (rooted) phylogenetic tree T, called a base tree, and adding arcs between arcs of T. The class of phylogenetic networks that can be obtained in this way is called tree-based networks and includes the prominent classes of tree-child and reticulation-visible networks. Initially defined for binary phylogenetic networks, tree-based networks naturally extend to arbitrary phylogenetic networks. In this paper, we generalise recent tree-based characterisations and associated proximity measures for binary phylogenetic networks to arbitrary phylogenetic networks. These characterisations are in terms of matchings in bipartite graphs, path partitions, and antichains. Some of the generalisations are straightforward to establish using the original approach, while others require a very different approach. Furthermore, for an arbitrary tree-based network N, we characterise the support trees of N, that is, the tree-based embeddings of N. We use this characterisation to give an explicit formula for the number of support trees of N when N is binary. This formula is written in terms of the components of a bipartite graph.


Phylogenetic network Tree-based network Nonbinary Matching Support tree Bipartite graph 

Mathematics Subject Classification

05C05 05C85 92D15 



We would like to thank Momoko Hayamizu for helpful comments concerning Theorem 8 and the two anonymous reviewers for further suggestions. JCP thanks the Biomathematics Research Centre of the University of Canterbury (and especially MS and CS) for hosting his visit, which led to this collaboration. We thank the (former) Allan Wilson Centre and the New Zealand Marsden Fund for funding support for this project.


  1. Anaya M, Anipchenko-Ulaj O, Ashfaq A, Chiu J, Kaiser M, Ohsawa MS, Owen M, Pavlechko E, St John K, Suleria S (2016) On determining if tree-based networks contain fixed trees. Bull Math Biol 78(5):961–969MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bang-Jensen J, Gutin G (2001) Digraphs: theory, algorithms and applications. Springer, BerlinzbMATHGoogle Scholar
  3. Cardona G, Llabrés M, Rosselló F, Valiente G (2008) A distance metric for a class of tree-sibling phylogenetic networks. Bioinformatics 24(13):1481–1488CrossRefGoogle Scholar
  4. Cardona G, Rossello F, Valiente G (2009) Comparison of tree-child phylogenetic networks. IEEE/ACM Trans Comput Biol Bioinform 6(4):552–569CrossRefGoogle Scholar
  5. Dagan T, Martin W (2006) The tree of one percent. Genome Biol 7(10):118CrossRefGoogle Scholar
  6. Doolittle WF, Bapteste E (2007) Pattern pluralism and the tree of life hypothesis. Proc Natl Acad Sci (USA) 104(7):2043–2049CrossRefGoogle Scholar
  7. Francis AR, Steel M (2015a) Tree-like reticulation networks—When do tree-like distances also support reticulate evolution? Math Biosci 259:12–19MathSciNetCrossRefzbMATHGoogle Scholar
  8. Francis AR, Steel M (2015b) Which phylogenetic networks are merely trees with additional arcs? Syst Biol 64(5):768–777CrossRefGoogle Scholar
  9. Francis A, Huber K, Moulton V (2018a) Tree-based unrooted phylogenetic networks. Bull Math Biol 80(2):404–416MathSciNetCrossRefzbMATHGoogle Scholar
  10. Francis A, Semple C, Steel M (2018b) New characterisations of tree-based networks and proximity measures. Adv Appl Math 93:93–107MathSciNetCrossRefzbMATHGoogle Scholar
  11. Hall P (1935) On representatives of subsets. J Lond Math Soc 10:26–30CrossRefzbMATHGoogle Scholar
  12. Hayamizu M (2016) On the existence of infinitely many universal tree-based networks. J Theor Biol 396:204–206MathSciNetCrossRefzbMATHGoogle Scholar
  13. Huson DH, Rupp R, Scornavacca C (2010) Phylogenetic networks: concepts, algorithms and applications. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  14. Jetten L (2015) Characterising tree-based phylogenetic networks (Karakterisatie van fylogenetische netwerken die een boom als basis hebben). Ph.D. Thesis, Delft University of TechnologyGoogle Scholar
  15. Jetten L, van Iersel L (2018) Nonbinary tree-based phylogenetic networks. IEEE/ACM Trans Comput Biol Bioinform 93:205–217CrossRefGoogle Scholar
  16. Linz S, St. John K, Semple C (2013) Counting trees in a phylogenetic network is #P-complete. SIAM J Discrete Math 42(4):1768–1776MathSciNetzbMATHGoogle Scholar
  17. Marcussen T, Sandve SR, Heier L, Spannagl M, Pfeifer M, [Consortium], TIWGS, Jakobsen K, Wulff BB, Steuernagel B, Mayer KF, Olsen O-A (2014) Ancient hybridizations among the ancestral genomes of bread wheat. Science 345(6194):1250092Google Scholar
  18. Morrison DA (2011) Introduction to phylogenetic networks. RJR Productions, UppsalaGoogle Scholar
  19. Semple C (2016) Phylogenetic networks with every embedded phylogenetic tree a base tree. Bull Math Biol 78(1):132MathSciNetCrossRefzbMATHGoogle Scholar
  20. van Iersel L (2013) Different topological restrictions of rooted phylogenetic networks: Which make biological sense? Accessed 15 Apr 2017
  21. Zhang L (2016) On tree-based phylogenetic networks. J Comput Biol 23(7):553–565MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of the Balearic IslandsPalmaSpain
  2. 2.School of Mathematics and StatisticsUniversity of CanterburyChristchurchNew Zealand

Personalised recommendations