When is a Phylogenetic Network Simply an Amalgamation of Two Trees?

Phylogenetic networks generalise phylogenetic (evolutionary) trees by allowing for the representation of reticulation (non-treelike) events. The structure of such networks is often viewed by the phylogenetic trees they embed. In this paper, we determine when a phylogenetic network [Formula: see text] has two phylogenetic tree embeddings which collectively contain all of the edges of [Formula: see text]. This determination leads to a polynomial-time algorithm for recognising such networks and an unexpected characterisation of the class of reticulation-visible networks.