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 $I$. We give a characterization of the families of real numbers that are realized by some weighted tree.

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