Labeling and folding multi-labeled trees

TL;DR

This paper extends a classic tree-labeling algorithm to multi-labeled trees, characterizing their structure through multiset partitions and linking them to labelable phylogenetic networks.

Contribution

It develops a new labeling algorithm for multi-labeled trees and connects these structures to labelable phylogenetic networks via multiset partitions.

Findings

Generalized labeling algorithm for multi-labeled trees

Characterization of multiset partitions from tree labelings

Bijection between labelable phylogenetic networks and multiset partitions

Abstract

In 1989 Erd\H{o}s and Sz\'ekely showed that there is a bijection between (i) the set of rooted trees with $n+1$ vertices whose leaves are bijectively labeled with the elements of $[\ell]=\{1,2,\dots,\ell\}$ for some $\ell \leq n$, and (ii) the set of partitions of $[n]=\{1,2,\dots,n\}$. They established this via a labeling algorithm based on the anti-lexicographic ordering of non-empty subsets of $[n]$ which extends the labeling of the leaves of a given tree to a labeling of all of the vertices of that tree. In this paper, we generalize their approach by developing a labeling algorithm for multi-labeled trees, that is, rooted trees whose leaves are labeled by positive integers but in which distinct leaves may have the same label. In particular, we show that certain orderings of the set of all finite, non-empty multisets of positive integers can be used to characterize partitions of a multiset that arise from labelings of multi-labeled trees. As an application, we show that the recently introduced class of labelable phylogenetic networks is precisely the class of phylogenetic networks that are stable relative to the so-called folding process on multi-labeled trees. We also give a bijection between the labelable phylogenetic networks with leaf-set $[n]$ and certain partitions of multisets.