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.

翻译：1989年，Erdős和Székely证明了存在一个双射关系，该关系连接了（i）具有$n+1$个顶点、叶子被双射标记为集合$[\ell]=\{1,2,\dots,\ell\}$中元素的根树（其中$\ell \leq n$），以及（ii）集合$[n]=\{1,2,\dots,n\}$的划分集合。他们通过一个基于$[n]$的非空子集的反字典序的标记算法建立了这一对应关系，该算法将给定树的叶子标记扩展到该树所有顶点的标记。在本文中，我们通过开发一种针对多标签树的标记算法来推广他们的方法，即那些叶子由正整数标记但不同叶子可能具有相同标签的根树。特别地，我们证明了正整数所有有限非空多重集合的某些排序可用于刻画由多标签树标记产生的多重集合划分。作为一个应用，我们证明了最近引入的可标记系统发育网络类恰好是相对于多标签树上的所谓折叠过程稳定的系统发育网络类。我们还给出了具有叶子集合$[n]$的可标记系统发育网络与某些多重集合划分之间的双射关系。