Efficiently computing runs on a trie

A maximal repetition, or run, in a string, is periodic substring whose smallest period at most half the length of substring. In this paper, we consider runs that correspond to path on trie, other words, rooted edge-labeled tree where each edge labeled with single symbol, and endpoints must be descendant/ancestor other. For trie n edges, show number less than n. We also an asymptotic lower bound maximum density tries: limn→∞⁡ρT(n)/n>0.9932348 ρT(n) edges. Furthermore, O(nlog⁡log⁡n) time O(n) space algorithm for finding all runs.