Paths in a random digital tree: limiting distributions

Abstract

We study a rule of growing a sequence {tn} of finite subtrees of an infinite m-ary tree T. Independent copies {ω (n)} of a Bernoulli-type process ω on m letters are used to trace out a sequence of paths in T. The tree tn is obtained by cutting each , at the first node such that at most σ paths out of , pass through it. Denote by Hn the length of the longest path, hn the length of the shortest path, and Ln the length of the randomly chosen path in tn. It is shown that, in probability, Hn – logan = O(1), hn – logb (n/log n) = 0(1), (or hn – logb (n/log log n) = O(1)), and that is asymptotically normal. The parameters a, b, c depend on the distribution of ω and, in case of a, also on σ. These estimates describe respectively the worst, the best and the typical case behavior of a ‘trie’ search algorithm for a dictionary-type information retrieval system, with σ being the capacity of a page.