International Journal of Mathematics and Mathematical Sciences
Volume 2005 (2005), Issue 23, Pages 3867-3882

Local extrema in random trees

Lane Clark

Department of Mathematics, College of Science, Southern Illinois University Carbondale, Carbondale 62901-4408, IL, USA

Received 23 March 2004; Revised 8 November 2005

Copyright © 2005 Lane Clark. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


The number of local maxima (resp., local minima) in a tree T𝒯n rooted at r[n] is denoted by Mr(T) (resp., by mr(T)). We find exact formulas as rational functions of n for the expectation and variance of M1(T) and mn(T) when T𝒯n is chosen randomly according to a uniform distribution. As a consequence, a.a.s. M1(T) and mn(T) belong to a relatively small interval when T𝒯n.