  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.

Populations and Evolution;Data Structures and Algorithms