Abstract: We review the definitions of two families of labeled graphs: both definitions generalize phylogenetic trees. Both families are posets and there are functions between them which form a Galois connection. To one of the sets there is naturally associated a polytope, and to the other, a metric space (which contains BHV space.) The connection extends to weighted networks. We show how to construct the nested polytopes, count their vertices (using associahedra), and find some facets.