A characterization of dissimilarity families of trees

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 $. In 2006 Levy, Yoshida and Pachter defined, for any positive-weighted tree ${\cal T}=(T,w)$ with $\{1,..., n\}$ as leaf set and any $i, j \in \{1,..., n\}$, the numbers $S_{i,j}$ to be $ \sum_{Y \in {\{1,..., n\} -\{i,j\} \choose k-2}} D_{i,j ,Y}({\cal T}) $; they proved that there exists a positive-weighted tree ${\cal T}' =(T',w')$ such that $D_{i,j}({\cal T}')=S_{i,j}$ for any $i,j \in \{1,..., n\}$ and that this new tree is, in some way, similar to the given one. In this paper, by using the $S_{i,j}$ defined by Levy, Yoshida and Pachter, we characterize families of real numbers parametrized by ${\{1,...,n\} \choose k}$ that are the families of $k$-weights of weighted trees with leaf set equal to $\{1,...., n\}$ and weights of the internal edges positive.