  We describe Galois connections which arise between two kinds of combinatorial
structures, both of which generalize trees with labelled leaves, and then apply
those connections to a family of polytopes.
  The graphs we study can be imbued with metric properties or associated to
vectors. Famous examples are the Billera-Holmes-Vogtmann metric space of
phylogenetic trees, and the Balanced Minimal Evolution polytopes of
phylogenetic trees described by Eickmeyer, Huggins, Pachter and Yoshida.
Recently the space of trees has been expanded to split networks by Devadoss and
Petti, while the definition of phylogenetic polytopes has been generalized to
encompass 1-nested phylogenetic networks, by Durell and Forcey. The first
Galois connection we describe is a reflection between the (unweighted) circular
split networks and the 1-nested phylogenetic networks. Another Galois
connection exists between certain metric versions of these structures.
Reflection between the purely combinatorial posets becomes a coreflection in
the geometric case.
  Our chief contributions here, beyond the discovery of the Galois connections,
are: a translation between approaches using PC-trees and networks, a new way to
look at weightings on networks, and a fuller characterization of faces of the
phylogenetic polytopes.
