Treelike families of multiweights

TL;DR

This paper characterizes the families of real numbers that can be realized as the k-weights of a weighted finite tree with given leaves, providing a mathematical framework for understanding tree structures from distance data.

Contribution

It offers a new characterization of families of real numbers that correspond to the k-weights of some weighted tree, advancing the understanding of tree reconstruction from partial distance data.

Findings

Provides necessary and sufficient conditions for a family to be realizable by a weighted tree.

Extends previous work on tree metrics to k-weights for arbitrary k.

Enhances methods for reconstructing tree structures from partial distance information.

Abstract

Let ${\cal T}=(T,w)$ be a weighted finite tree with leaves $1,..., n$. For any $I :=\{i_1,..., i_k \} \subset \{1,...,n\}$, let $D_I ({\cal T})$ be the weight of the minimal subtree of $T$ connecting $i_1,..., i_k$; the $D_{I} ({\cal T})$ are called $k$-weights of ${\cal T}$. Given a family of real numbers parametrized by the $k$-subsets of $\{1,..., n\}$, $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$, we say that a weighted tree ${\cal T}=(T,w)$ with leaves $1,..., n$ realizes the family if $D_I({\cal T})=D_I$ for any $ I $. We give a characterization of the families of real numbers that are realized by some weighted tree.