  While every rooted binary phylogenetic tree is determined by its set of
displayed rooted triples, such a result does not hold for an arbitrary rooted
binary phylogenetic network. In particular, there exist two non-isomorphic
rooted binary temporal normal networks that display the same set of rooted
triples. Moreover, without any structural constraint on the rooted phylogenetic
networks under consideration, similarly negative results have also been
established for binets and trinets which are rooted subnetworks on two and
three leaves, respectively. Hence, in general, piecing together a rooted
phylogenetic network from such a set of small building blocks appears
insurmountable. In contrast to these results, in this paper, we show that a
rooted binary normal network is determined by its sets of displayed
caterpillars (particular type of subtrees) on three and four leaves. The proof
is constructive and realises a polynomial-time algorithm that takes the sets of
caterpillars on three and four leaves displayed by a rooted binary normal
network and, up to isomorphism, reconstructs this network.
