The properties of random trees

Abstract

Consider an s-ary tree (in which every node has no more than s children). Each node holds a single datum, including a key. These are the occupied, internal, or closed nodes of the data structure. Augment the tree, following D. E. Knuth, by adding a set of unoccupied, external, open, or free nodes, so that every internal node now has just s children and every external node has no children. We assume that there is an unambiguous rule, depending only on the key values at the internal nodes of the tree, whereby a new datum, with a new key value, will be inserted at one of the external nodes; this node then becomes internal and acquires s new external nodes as children. We further assume that the rule and the statistical distribution of data are such that every external node has equal probability of being selected for insertion of a new datum, at every stage. Various statistics of such trees are now obtained explicitly, in a systematic manner which may be extended to higher moments. The principal result is that the average level of both internal and external nodes in a given tree is asymptotic in probability to [ s (s−1)]log m as m → ∞, where m is the number of internal nodes in the tree. Since the corresponding average level for a k-level fully balanced tree [with m = (s k−1) (s−1) ] is asymptotic to k ≈ log s m = ( l log s )log m as m → ∞, we conclude that, unless the distribution of data is far from the rather plausible assumption made here, it is highly improbable that the considerable cost of rebalancing trees when constructing data-bases will ever be justified in practice.