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.
Categories: math.CO