  Binets and trinets are phylogenetic networks with two and three leaves,
respectively. Here we consider the problem of deciding if there exists a binary
level-1 phylogenetic network displaying a given set $\mathcal{T}$ of binary
binets or trinets over a set $X$ of taxa, and constructing such a network
whenever it exists. We show that this is NP-hard for trinets but
polynomial-time solvable for binets. Moreover, we show that the problem is
still polynomial-time solvable for inputs consisting of binets and trinets as
long as the cycles in the trinets have size three. Finally, we present an
$O(3^{|X|} poly(|X|))$ time algorithm for general sets of binets and trinets.
The latter two algorithms generalise to instances containing level-1 networks
with arbitrarily many leaves, and thus provide some of the first supernetwork
algorithms for computing networks from a set of rooted phylogenetic networks.
