Treelike families of multiweights

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 $. Weighted graphs have applications in several disciplines, such as biology, archaeology, engineering, computer science, in fact, they can represent hydraulic webs, railway webs, computer networks...; moreover, in biology, weighted trees are used to represent the evolution of the species. In this paper we give a characterization of the families of real numbers parametrized by the $k$-subsets of some set that are realized by some weighted tree.