  In the context of reconstructing phylogenetic networks from a collection of
phylogenetic trees, several characterisations and subsequently algorithms have
been established to reconstruct a phylogenetic network that collectively embeds
all trees in the input in some minimum way. For many instances however, the
resulting network also embeds additional phylogenetic trees that are not part
of the input. However, little is known about these inferred trees. In this
paper, we explore the relationships among all phylogenetic trees that are
embedded in a given phylogenetic network. First, we investigate some
combinatorial properties of the collection P of all rooted binary phylogenetic
trees that are embedded in a rooted binary phylogenetic network N. To this end,
we associated a particular graph G, which we call rSPR graph, with the elements
in P and show that, if |P|=2^k, where k is the number of vertices with
in-degree two in N, then G has a Hamiltonian cycle. Second, by exploiting rSPR
graphs and properties of hypercubes, we turn to the well-studied class of
rooted binary level-1 networks and give necessary and sufficient conditions for
when a set of rooted binary phylogenetic trees can be embedded in a level-1
network without inferring any additional trees. Lastly, we show how these
conditions translate into a polynomial-time algorithm to reconstruct such a
network if it exists.
