  Rooted phylogenetic networks are often constructed by combining trees,
clusters, triplets or characters into a single network that in some
well-defined sense simultaneously represents them all. We review these four
models and investigate how they are related. In general, the model chosen
influences the minimum number of reticulation events required. However, when
one obtains the input data from two binary trees, we show that the minimum
number of reticulations is independent of the model. The number of
reticulations necessary to represent the trees, triplets, clusters (in the
softwired sense) and characters (with unrestricted multiple crossover
recombination) are all equal. Furthermore, we show that these results also hold
when not the number of reticulations but the level of the constructed network
is minimised. We use these unification results to settle several complexity
questions that have been open in the field for some time. We also give explicit
examples to show that already for data obtained from three binary trees the
models begin to diverge.
