  It is an open question whether there exists a polynomial-time algorithm for
computing the rotation distances between pairs of extended ordered binary
trees. The problem of computing the rotation distance between an arbitrary pair
of trees, (S, T), can be efficiently reduced to the problem of computing the
rotation distance between a difficult pair of trees (S', T'), where there is no
known first step which is guaranteed to be the beginning of a minimal length
path. Of interest, therefore, is how to sample such difficult pairs of trees of
a fixed size. We show that it is possible to do so efficiently, and present
such an algorithm that runs in time $O(n^4)$.
