A Greedy Algorithm Estimating the Height of Random Trees

The behaviour of a greedy algorithm which estimates the height of a random labelled rooted tree is studied. A self-similarity argument is used to characterize the limit distribution of the length H of the path found by such an algorithm in a random rooted tree as the unique solution of an integral equation. Furthermore, it is shown that lim n!1 E H p n = p 2 2 p 2 ln(3 + 2 p 2) = 2 : 353139::: ; i.e. the expected length of the path constructed by the algorithm is roughly 93% of the expected height of a random rooted tree.