General Restricted Rotation Distance

The restricted rotation distance d_{R}(S,T) between two trees S, T of n vertices is the minimum number of rotations by which S can be transformed into T, where rotations can only take place at the root of the tree, or at the right child of the root. A sharp upper bound d_{R}(S,T)\leq 4n-8 is known, based on the word metric of Thompson's group. We refine this bound to d_{R}(S,T)\leq 4n-8-\rho_{S} \rho_{T}, where \rho_{S} and \rho_{T} are the numbers of vertices in the rightmost vertex chains of the two trees, providing a very simple transformation algorithm based on elementary properties of trees. We also generalize the concept of restricted rotation by allowing rotations at the highest k levels of the tree, and study the new distance for k=2,3.