Labeling and folding multi-labeled trees

Abstract

In 1989 Erdős and Székely 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.

Figures (6)

• Figure 1: (a) A semi-labeled tree with $\ell=5$ and $n=9$. (b) The semi-labeled tree in (a) with vertices bijectively labeled with the elements of $[9]$. Taking for each non-leaf vertex $v$ the subset of $[9]$ labeling the children of $v$, yields the partition $\{\{1,3\},\{2,4\},\{5,6\},\{8\},\{7,9\}\}$ of $[9]$. (c) A multi-labeled tree. (d) The phylogenetic network obtained by folding the multi-labeled tree in (c).
• Figure 2: (a) A leaf-labeled network $(\mathcal{N},\lambda)$. (b) The leaf-labeled network $(\mathcal{N}_u,\lambda_u)$ for the vertex $u$ in (a). (c) The fully-labeled network produced by Algorithm \ref{['alg:label:network']} from the leaf-labeled network in (a) using as $\preceq$ the lexicographic ordering.
• Figure 3: (a) A leaf-labeled tree $(\mathcal{T},\lambda)$. (b) The full labeling $\phi=\phi_{(\lambda,\preceq)}$ of the leaf-labeled tree in (a) produced by Algorithm \ref{['alg:label:network']} using as $\preceq$ the lexicographic ordering. Interior vertices $u,v$ with the same label have isomorphic leaf-labeled trees $(\mathcal{T}_u,\lambda_u)$ and $(\mathcal{T}_v,\lambda_v)$ (cf. Lemma \ref{['lem:label:tree:isomorphic']}).
• Figure 4: Two non-isomorphic leaf-labeled trees $(\mathcal{T}_1,\lambda_1)$ and $(\mathcal{T}_2,\lambda_2)$ with $\Pi_{(\mathcal{T}_1,\lambda_1,\preceq)} = \Pi_{(\mathcal{T}_2,\lambda_2,\preceq)} = \{\{1,2\},\{1,1,3\}\}$ (independently of the choice of $\preceq$ in Algorithm \ref{['alg:label:network']}).
• Figure 5: (a) A leaf-labeled tree illustrating that the lexicographic ordering of $\mathbb{M}$ is not labeling-consistent. (b) The structure of the leaf-labeled tree used in the proof of Lemma \ref{['lem:char:labeling:consistent']}. Each triangle represents a set of leaves that are bijectively labeled by the elements of the multiset below the triangle.

Theorems & Definitions (16)

• Lemma 2.1
• Theorem 2.2
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Remark 3.4
• Lemma 4.1
• Corollary 4.2
• Theorem 4.3
• Lemma 5.1