The -Cophenetic Metric for Phylogenetic Trees As an Interleaving Distance

  • Elizabeth MunchEmail author
  • Anastasios Stefanou
Part of the Association for Women in Mathematics Series book series (AWMS, volume 17)


There are many metrics available to compare phylogenetic trees since this is a fundamental task in computational biology. In this paper, we focus on one such metric, the -cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with n labeled leaves as a point in \(\mathbb {R}^{n(n+1)/2}\) known as the cophenetic vector, then comparing the two resulting Euclidean points using the distance. Meanwhile, the interleaving distance is a formal categorical construction generalized from the definition of Chazal et al., originally introduced to compare persistence modules arising from the field of topological data analysis. We show that the -cophenetic metric is an example of an interleaving distance. To do this, we define phylogenetic trees as a category of merge trees with some additional structure, namely, labelings on the leaves plus a requirement that morphisms respect these labels. Then we can use the definition of a flow on this category to give an interleaving distance. Finally, we show that, because of the additional structure given by the categories defined, the map sending a labeled merge tree to the cophenetic vector is, in fact, an isometric embedding, thus proving that the -cophenetic metric is an interleaving distance.


Topological data analysis Labeled merge tree Phylogenetic tree Interleaving distance Category with a flow 



The authors gratefully thank two anonymous reviewers whose feedback substantially increased the quality of the paper. The work of EM was supported in part by NSF Grant Nos. DMS-1800446 and CMMI-1800466. AS was partially supported both by the National Science Foundation through grant NSF-CCF-1740761 TRIPODS TGDA@OSU and by the Mathematical Biosciences Institute at the Ohio State University.


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Copyright information

© The Author(s) and the Association for Women in Mathematics 2019

Authors and Affiliations

  1. 1.Department of Computational Mathematics, Science and EngineeringDepartment of Mathematics, Michigan State UniversityEast LansingUSA
  2. 2.Mathematical Biosciences Institute, Department of MathematicsThe Ohio State UniversityColumbusUSA

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