  Rotation distance between rooted binary trees is the minimum number of simple
rotations needed to transform one tree into the other. Computing the rotation
distance between a pair of rooted trees can be quickly reduced in cases where
there is a common edge between the trees, or where a single rotation introduces
a common edge. Tree pairs which do not have such a reduction are difficult tree
pairs, where there is no generally known first step. Here, we describe efforts
to count and estimate the number of such difficult tree pairs, and find that
the fraction decreases exponentially fast toward zero. We also describe how
knowing the number of distinct instances of the rotation distance problem is a
helpful factor in making the computations more feasible.
