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 T r(N ) denote the set of all trees displayed by the network N . In general, there may be many other networks M such that T r(M ) = T r(N ). A network is regular if it is isomorphic with its cover digraph. If N is regular and D is a collection of trees displayed by N , this paper studies some procedures to try to reconstruct N

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... en D. If the input is D = T r(N ), one procedure is described which will reconstruct N . Hence if N and M are regular networks and T r(N ) = T r(M ), it follows that N = M , proving that a regular network is uniquely determined by its displayed trees. If D is a (usually very much smaller) collection of displayed trees that satisfies certain hypotheses, modifications of the procedure will still reconstruct N given D.