A phylogenetic network is a directed acyclic graph that visualises
an evolutionary history containing so-called reticulations such
as recombinations, hybridisations or lateral gene transfers. Here we consider
the construction of a simplest possible phylogenetic network consistent
with an input set T, where T contains at least one phylogenetic
tree on three leaves (a triplet) for each combination of three taxa. To
quantify the complexity of a network we consider both the total number
of reticulations and the number of reticulations per biconnected component,
called the level of the network. We give polynomial-time algorithms
for constructing a level-1 respectively a level-2 network that contains a
minimum number of reticulations and is consistent with T (if such a
network exists). In addition, we show that if T is precisely equal to the
set of triplets consistent with some network, then we can construct such
a network, which minimises both the level and the total number of reticulations,
in time O(|T|^{k+1}), if k is a fixed upper bound on the level
