Treelike families of multiweights

Abstract

Let $\mathcal{T}=(T,w)$ be a weighted finite tree with leaves $1,...,n$. For any $I:=\{i_1,...,i_k\}\subset \{1,...,n\}$, let $D_I(\mathcal{T})$ be the weight of the minimal subtree of $T$ connecting $i_1,...,i_k$; the $D_I(\mathcal{T})$ are called $k$-weights of $\mathcal{T}$. Given a family of real numbers parametrized by the $k$-subsets of $\{1,...,n\}$, $\{D_I{\}}_{I\in \left(\genfrac{}{}{0ex}{}{\{1,...,n\}}{k}\right)}$, we say that a weighted tree $\mathcal{T}=(T,w)$ with leaves $1,...,n$ realizes the family if $D_I(\mathcal{T})=D_I$ for any $I$. We give a characterization of the families of real numbers that are realized by some weighted tree.

17 pages