  We introduce and study the model of simply generated non-crossing partitions,
which are, roughly speaking, chosen at random according to a sequence of
weights. This framework encompasses the particular case of uniform non-crossing
partitions with constraints on their block sizes. Our main tool is a bijection
between non-crossing partitions and plane trees, which maps such simply
generated non-crossing partitions into simply generated trees so that blocks of
size $k$ are in correspondence with vertices of outdegree $k$. This allows us
to obtain limit theorems concerning the block structure of simply generated
non-crossing partitions. We apply our results in free probability by giving a
simple formula relating the maximum of the support of a compactly supported
probability measure on the real line in term of its free cumulants.
