  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 $\ell^\infty$-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 $\ell^\infty$ 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 $\ell^\infty$-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
$\ell^\infty$-cophenetic metric is, in fact, an interleaving distance.
