Families of multiweights and pseudostars

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${\cal T}=(T,w)$ with leaves $1,\mathrm{.}\mathrm{.}\mathrm{.},n$ realizes the family if $D_I(\mathcal{T})=D_I$ for any $I$. In [P-S] Pachter and Speyer proved that, if $3\le k\le (n+1)\mathrm{/}2$ and $\{D_I{\}}_{I\in \left(\genfrac{}{}{0px}{}{\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}}{k}\right)}$ is a family of positive real numbers, then there exists at most one positive-weighted essential tree $\mathcal{T}$ with leaves $1,\mathrm{.}\mathrm{.}\mathrm{.},n$ that realizes the family (where “essential” means that there are no vertices of degree $2$). We say that a tree $P$ is a pseudostar of kind $(n,k)$ if the cardinality of the leaf set is $n$ and any edge of $P$ divides the leaf set into two sets such that at least one of them has cardinality $\ge k$. Here we show that, if $3\le k\le n-1$ and ${D_I}{I \in {{1,…,n} \choose k}}$ is a family of real numbers realized by some weighted tree, then there is exactly one weighted essential pseudostar $\mathcal{P}=(P,w)$ of kind $(n,k)$ with leaves $1,\mathrm{.}\mathrm{.}\mathrm{.},n$ and without internal edges of weight $0$, that realizes the family; moreover we describe how any other weighted tree realizing the family can be obtained from $\mathcal{P}$. Finally we examine the range of the total weight of the weighted trees realizing a fixed family.