A characterization of dissimilarity families of trees

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 $D_I(\mathcal{T})$ are called $k$-weights of $\mathcal{T}$. 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${\cal T}=(T,w)$withleaves$1,…, n$realizesthefamilyif$D_I({\cal T})=D_I$forany$ I $. In 2006 Levy, Yoshida and Pachter defined, for any positive-weighted tree $\mathcal{T}=(T,w)$ with $\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ as leaf set and any $i, j \in {1,…, n}$,thenumbers$S{i,j}$tobe$ \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 ${\mathcal{T}}^{\mathrm{\prime }}=(T^{\mathrm{\prime }},w^{\mathrm{\prime }})$ such that $D_{i,j}({\mathcal{T}}^{\mathrm{\prime }})=S_{i,j}$ 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 ${{1,…,n} \choose k}$thatarethefamiliesof$k$-weights of weighted trees with leaf set equal to $\{1,\mathrm{.}\mathrm{.}\mathrm{.}\mathrm{.},n\}$ and weights of the internal edges positive.