Rotation Distance , Triangulations , and Hyperbolic Geometry Daniel

A rotation in a binary tree is a local restructuring that changes thetree into another tree. Rotations are useful in the design of tree-based data struc-tures. The rotation distance between a pair of trees is the minimum number ofrotations needed to convert one tree into the other. In this paper we estab-lish a tight bound of 2n 6 on the maximum rotation distance between twon-node trees for all large n. The hard and novel part of the proof is the lowerbound, which makes use of volumetric arguments in hyperbolic 3-space. Ourproof also gives a tight bound on the minimum number of tetrahedra neededto dissect a polyhedron in the worst case and reveals connections among binarytrees, triangulations, polyhedra, and hyperbolic geometry. COMPUTER SCIENCE DEPARTMENT, CARNEGIE MELLON UNIVERSITY, PITTSBURGH, PENNSYLVA-NIA 15213 COMPUTER SCIENCE DEPARTMENT, PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY 08544 AT&T; BELL LABORATORIES, MURRAY HILL, NEW JERSEY 07974 MATHEMATICS DEPARTMENT, PRINCETON UNIVERSITY, PRINCETON, NEW JERSEY 08544 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use