Families of multiweights and pseudostars

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 $I$. In [P-S] Pachter and Speyer proved that, if $3 \leq k \leq (n+1)/2$ and $\{D_I\}_{I \in {\{1,...,n\} \choose k}}$ is a family of positive real numbers, then there exists at most one positive-weighted essential tree ${\cal T}$ with leaves $1,...,n$ 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 $\geq k$. Here we show that, if $3 \leq k \leq n-1$ 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 ${\cal P}=(P,w)$ of kind $(n,k)$ with leaves $1,...,n$ 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 ${\cal P}$. Finally we examine the range of the total weight of the weighted trees realizing a fixed family.

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