# The ℓ^∞-Cophenetic Metric for Phylogenetic Trees as an Interleaving Distance

arXiv: https://arxiv.org/abs/1803.07609 · HTML: https://mlsanity.com/paper/1803.07609/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Feb 2018.

## Built on by (most cited first, 1 of 1)

- [The Universal ℓp-Metric on Merge Trees](https://mlsanity.com/paper/2112.12165/) (2021)

## Abstract (arXiv)

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 ℓ^∞-cophenetic metric introduced by Cardona et al. This metric works by representing a phylogenetic tree with n labeled leaves as a point in ℝn(n+1)/2 known as the cophenetic vector, then comparing the two resulting Euclidean points using the ℓ^∞ 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 ℓ^∞-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 ℓ^∞-cophenetic metric is, in fact, an interleaving distance.