Cactus tree type models for families of bisections of a set

A bisection of a set U is a partition of U into two nonempty parts. Two bisections are parallel if they collectively partition U into three parts. A natural model for a family of pairwise parallel bisections is a tree. Given a bisection family F and a family T of pairwise parallel bisections, we suggest decomposition/composition tools for modeling F based on T . We introduce plant models resulting from such compositions. As an application, we obtain a simple characterization of bisection families that can be modeled by a cactus-tree (i.e., a tree-of-edges-and-cycles) and its variants. We use this characterization to derive several related results.