  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 {\em 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 necessary uniquely determine or {\em 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
{\em 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 {\em 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.
