Treelike families of multiweights

Let $\mathcal{T}=(T,w)$ be a weighted finite tree with leaves $1,\mathrm{.}\mathrm{.}\mathrm{.},n$. For any $I:=\{i_1,\mathrm{.}\mathrm{.}\mathrm{.},i_k\}\subset \{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$, let $D_I(\mathcal{T})$ be the weight of the minimal subtree of $T$ connecting $i_1,\mathrm{.}\mathrm{.}\mathrm{.},i_k$; the $D_{I} ({\cal T})$arecalled$k$-weightsof${\cal T}$. Given a family of real numbers parametrized by the $k$-subsets of $\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$, ${D_I}_{I \in {{1,…,n} \choose k}}$,wesaythataweightedtree${\cal T}=(T,w)$ with leaves $1,\mathrm{.}\mathrm{.}\mathrm{.},n$ 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.