  Motivated by the study of pattern avoidance in the context of permutations
and ordered partitions, we consider the enumeration of weak-ordering chains
obtained as leaves of certain restricted rooted trees. A tree of order $n$ is
generated by inserting a new variable into each node at every step. A node
becomes a leaf either after $n$ steps or when a certain stopping condition is
met. In this paper we focus on conditions of size 2 ($x=y$, $x<y$, or $x\le y$)
and several conditions of size 3. Some of the cases considered here lead to the
study of descent statistics of certain `almost' pattern-avoiding permutations.
