On the Markov Chain for the Move-to-Root Rule for Binary Search Trees

The move-to-root (MTR) heuristic is a self-organizing rule that attempts to keep a binary search tree in near-optimal form. It is a tree analogue of the move-to-front (MTF) scheme for self-organizing lists. Both heuristics can be modeled as Markov chains. We show that the MTR chain can be derived by lumping the MTF chain and give exact formulas for the transition probabilities and stationary distribution for MTR. We also derive the eigenvalues and their multiplicities for MTR. 1. Introduction and summary. There has been much interest in recent years in self-organizing search methods. Hester and Hirschberg (1985) survey the field. Hendricks (1989) is a good introduction with numerous applications and open problems. Although most research in this area has been devoted to sequential search techniques for linear lists, a growing body of work addresses heuristics for other data structures. In particular, the binary search tree is a very common and important structure that exploits the ordering of records to achieve faster search time. Records are stored at the nodes of a tree in such a way that a traversal of the tree produces the records in their linear order. A binary tree is a finite tree with at most two "children" for each node and in which each child is distinguished as either a left or right child. By defining an empty binary tree as a binary tree with no nodes, we can give a useful recursive definition: a binary tree either is empty or is a node with left and right subtrees, each of which is a binary tree.