On the rotation distance between binary trees

We develop combinatorial methods for computing the rotation distance between binary trees, i.e., equivalently, the flip distance between triangulations of a polygon. As an application, we prove that, for each n, there exist size n trees at distance 2n − O( √ n). If T, T ′ are finite binary rooted trees, one says that T ′ is obtained from T by one rotation if T ′ coincides with T except in the neighbourhood of some inner node where the branching patterns respectively are ... . . . ... ... and ...