Phylogenetic networks are a generalization of phylogenetic trees that are used in biology to represent reticulate or non‐treelike evolution. Recently, several algorithms have been developed which aim to construct phylogenetic networks from biological data using triplets, i.e. binary phylogenetic trees on 3‐element subsets of a given set of species. However, a fundamental problem with this approach is that the triplets displayed by a phylogenetic network do not necessarily uniquely determine or encode the network. Here we propose an alternative approach to encoding and constructing phylogenetic networks, which uses phylogenetic networks on 3‐element subsets of a set, or trinets, rather than triplets. More specifically, we show that for a special, well‐studied type of phylogenetic network called a 1‐nested network, the trinets displayed by a 1‐nested network always encode the network. We also present an efficient algorithm for deciding whether a dense set of trinets (i.e. one that contains a trinet on every 3‐element subset of a set) can be displayed by a 1‐nested network or not and, if so, constructs that network. In addition, we discuss some potential new directions that this new approach opens up for constructing and comparing phylogenetic networks.
