  A rooted acyclic digraph N with labelled leaves displays a tree T when there
exists a way to select a unique parent of each hybrid vertex resulting in the
tree T. Let Tr(N) denote the set of all trees displayed by the network N. In
general, there may be many other networks M such that Tr(M) = Tr(N). A network
is regular if it is isomorphic with its cover digraph. This paper shows that if
N is regular, there is a procedure to reconstruct N given Tr(N). Hence if N and
M are regular networks and Tr(N) = Tr(M), it follows that N = M, proving that a
regular network is uniquely determined by its displayed trees.
