We describe Galois connections which arise between two kinds of combinatorial structures, both of which generalize trees with labeled leaves, and then apply those connections to a family of polytopes. The graphs we study can be imbued with metric properties or associated with 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.

我们描述了在两种组合结构之间产生的Galois连接，这两种组合结构都将带有标记叶的树泛化，然后将这些连接应用于多面体家族。我们研究的图形可以充满度量属性或与向量相关联。著名的例子是系统发育树的Billera–Holmes–Vogtmann度量空间，以及Eickmeyer，Huggins，Pachter和Yoshida描述的系统发育树的平衡最小进化多态性。最近，Devadoss和Petti已将树木的空间扩展为分裂网络，而Durell和Forcey已将系统发育多面体的定义概括为包含1个嵌套的系统发育网络。我们描述的第一个Galois连接是（未加权的）圆形分裂网络与1嵌套系统发生网络之间的反映。这些结构的某些度量标准版本之间存在另一个Galois连接。在几何情况下，纯组合式样球之间的反射变成了一个核心弯曲。除了发现Galois连接之外，我们在这里的主要贡献是：使用PC树和网络的方法之间的转换，一种查看网络权重的新方法以及对系统发育多面体的面孔进行更全面的表征。