Let (T = (V, E), P, ω) be an instance of W-MTO where T is a rooted tree.
A (rooted) phylogenetic tree is a (rooted) phylogenetic network for which it is not possible to delete any edge without producing a graph that is not connected.
A rooted binary phylogenetic X-tree T is a rooted tree whose root has degree two while all other interior vertices have degree three and whose leaf set is X.
Another homogeneous tree is a rooted Cayley tree T C,N --the root has N neighbors, and the other vertices have N + 1 neighbors.
A phylogenetic X-network N = (V, A, r, X) is a rooted acyclic digraph G = (V, A) with root r such that there is a one-to-one map ϕ : X → V whose image contains all vertices v such that either (i) v is a leaf; or (ii) v = r; or (iii) v has indegree 1 and outdegree 1.
In the special case when (R_{i}) is a rooted star (K_{1,a_{i}}) with the center of the star as its root (that is, (e(v_{i})=1)), we will simplify the notation by replacing (R_{i}) by the number (a_{i}).
