Treelike Families of Multiweights

Abstract

Let T = (T,w) be a weighted finite tree with leaves 1, ..., n. For any I := {i₁, ..., i_k} ⊂ {1, ..., n}, let D_I (T ) be the weight of the minimal subtree of T connecting i₁, ..., i_k; the D_I (T ) are called k-weights of T . Given a family of real numbers parametrized by the k-subsets of $\left\{1,\dots, n\right\},{\left\{{D}_I\right\}}_{I\in \left(\underset{k}{\left\{1,\dots, n\right\}}\right)},$ we say that a weighted tree T = (T,w) with leaves 1, ..., n realizes the family if D_I (T ) = D_I for any I. Weighted graphs have applications in several disciplines, such as biology, archaeology, engineering, computer science, in fact, they can represent hydraulic webs, railway webs, computer networks...; moreover, in biology, weighted trees are used to represent the evolution of the species. In this paper we give a characterization of the families of real numbers parametrized by the k-subsets of some set that are realized by some weighted tree.