  Reticulate evolution can be modelled using phylogenetic networks. Tree-based
networks, which are one of the more general classes of phylogenetic networks,
have recently gained eminence for its ability to represent evolutionary
histories with an underlying tree structure. To better understand tree-based
networks, numerous characterizations have been proposed, based on tree
embeddings, matchings, and arc partitions. Here, we build a bridge between two
arc partition characterizations, namely maximal fence decompositions and cherry
covers. Results on cherry covers have been found for general phylogenetic
networks. We first show that the number of cherry covers is the same as the
number of support trees (underlying tree structure of tree-based networks) for
a given semibinary network. Maximal fence decompositions have only been defined
thus far for binary networks (constraints on vertex degrees). We remedy this by
generalizing fence decompositions to non-binary networks, and using this, we
characterize semi-binary tree-based networks in terms of forbidden structures.
Furthermore, we give an explicit enumeration of cherry covers of semi-binary
networks, by studying its fence decomposition. Finally, we prove that it is
possible to characterize semi-binary tree-child networks, a subclass of
tree-based networks, in terms of the number of their cherry covers.
