Tight Upper and Lower Bounds on the Path Length of Binary Trees

The external path length of a tree T is the sum of the lengths of the paths from the root to each external node. The maximal path length diierence, , is the diierence between the length of the longest and shortest such path. We prove tight lower and upper bounds on the external path length of binary trees with N external nodes and prescribed maximal path length diierence. In particular, we give an upper bound that, for each value of , can be exactly achieved for innnitely many values of N. This improves on the previously known upper bound that could only be achieved up to a factor proportional to N. We also present, as a preliminary result, an elementary proof of the known upper bound. We nally prove a lower bound which can be exactly achieved for each value of N and N=2. AMS(MOS) subject classiications. 68P05, 68P10 1. Introduction. Binary trees constitute the most important and widely used data structure for the storage and retrieval of information. The cost of accessing information stored in a node is proportional to the distance of the node from the root. An important measure of eeciency, often considered with respect to a binary tree T, is its external path length External(T), that is the sum of the distances of the external nodes from the root as this is related to the average cost of an unsuccessful search in the tree T. Moreover, the external path length of T is related to the internal path length Internal(T) that is the sum of the distances of the internal nodes from the root and corresponds to the average cost of a successful search in the tree T. It is well known that the external path length of a binary tree with N external nodes is (N lg N) in the best case while in the worst case can be as bad as (N 2). The large gap between the best and the worst case motivates the study of this important quantity when some additional information about the tree is available. Nievergelt and Wong 7] proposed an upper bound for the external path length of a tree T in term of the number of external nodes and the maximum weight balance of all its subtrees. More recently, Klein and Wood 5] obtained an upper bound that requires much less information about the tree. Namely, they derived …