  This extended abstract is dedicated to the analysis of the height of
non-plane unlabelled rooted binary trees. The height of such a tree chosen
uniformly among those of size $n$ is proved to have a limiting theta
distribution, both in a central and local sense. Moderate as well as large
deviations estimates are also derived. The proofs rely on the analysis (in the
complex plane) of generating functions associated with trees of bounded height.
