K-Restricted rotation distance between binary trees

Abstract

The restricted rotation distance d R ( 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 d R ( S , T ) ⩽ 4 n − 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 d R ( S , T ) ⩽ 4 n − 8 − ρ S − ρ T , where ρ S and ρ T 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.