On dissimilarity vectors of (not necessarily positive) weighted trees

TL;DR

This paper characterizes the sets of real numbers that can represent the k-weights of a weighted tree with n leaves, extending the understanding of dissimilarity vectors beyond positive weights.

Contribution

It provides a characterization of the sets of real numbers that correspond to the k-weights of possibly non-positive weighted trees with a fixed number of leaves.

Findings

Provides necessary and sufficient conditions for a set of numbers to be k-weights of a weighted tree.

Extends previous results to include trees with non-positive edge weights.

Offers a framework for reconstructing trees from dissimilarity data.

Abstract

Let T be a (not necessarily positive) weighted tree with n leaves numbered by the set {1,...,n}. Define the k-weights of the tree D_{i_1,....,i_k}(T) as the sum of the lengths of the edges of the minimal subtree connecting i_1,....,i_k. We will call such numbers "k-weights" of the tree. In this paper, we characterize the sets of real numbers indexed by the subsets of any cardinality >= 2 of a n-set to be the weights of a tree with n leaves.