k-Restricted rotation distance between binary trees

The restricted rotation distance dR(S,T) between two binary trees S, T of n vertices is the minimum number of rotations to transform S into T, where rotations take place at the root of S, or at the right child of the root. A sharp upper bound dR(S,T)⩽4n−8 is known, based on group theory [S. Cleary, J. Taback, Bounding restricted rotation distance, Information Processing Letters 88 (5) (2003) 251–256]. We refine this bound to a sharp dR(S,T)⩽4n−8−ρS−ρT, where ρS and ρT are the numbers of vertices in the rightmost vertex chains of the two trees, using an elementary transformation algorithm. We then generalize the concept to k-restricted rotation, by allowing rotations to take place at all the vertices of the highest k levels of the tree, and study the new distance for k=2. The case k⩾3 is essentially open.