  We study the fundamental question of how likely it is that two randomly
chosen trees are isomorphic to each other for different models of random trees.
We show that the probability decays exponentially for rooted labeled trees as
well as for Galton--Watson trees with bounded degrees but that this is not true
for plane trees, thus providing a counterexample in the general case of
Galton--Watson trees without degree restrictions. We also derive limiting
distributions for some related parameters: the number of vertices of given
degrees in pairs of labeled trees conditioned on being isomorphic, thus showing
that they have a different shape than usual labeled trees, as well as the
number of labelings and plane representations of P\'olya trees. The results
rely on both probabilistic and analytic tools.
