that has reticulation number zero. We say that a network $N$ on $\mathcal{X}$ *displays* a tree $T$ if $T$ can be obtained from $N$ by performing a series of node and edge deletions and eventually by suppressing nodes with both indegree and outdegree equal to 1. We assume without loss of generality that each reticulation has outdegree at least one. Consequently, each leaf has indegree one. We say that a network is *binary* if every reticulation node has indegree 2 and outdegree 1 and every tree node that is not a leaf has outdegree 2. Proper subsets of $\mathcal{X}$ are called *clusters*, and a cluster $C$ is a *singleton* if $|C|=1$. We say that an edge $(u,v)$ of a tree *represents* a cluster $C \subset {{\mathcal X}}$ if $C$ is the set of leaf descendants of $v$. A tree $T$ represents a cluster $C$ if it contains an edge that represents $C$. It is well-known that the set of clusters represented by a tree is a laminar set, and uniquely defines that tree. We say that a network $N$ represents a cluster $C \subset \mathcal{X}$ “in the hardwired sense” if there exists a tree edge $(u,v)$ of $N$ such that 