  The classes of tree permutations and forest permutations were defined by Acan
and Hitczenko (2016). We study random permutations of a given length from these
classes, and in particular the number of occurrences of a fixed pattern in one
of these random permutations. The main results show that the distributions of
these numbers are asymptotically normal.
  The proof uses representations of random tree and forest permutations that
enable us to express the number of occurrences of a pattern by a type of
$U$-statistics; we then use general limit theorems for the latter.
