Profiles of Random Trees: Correlation and Width of Ran- Dom Recursive Trees and Binary Search Trees

The levels of trees are nodes with the same distance to the root. We derive asymptotic approximations to the correlation coefficients of two level sizes in random recursive trees and binary search trees, which undergo sharp sign-changes when one level is fixed and the other one is varying. We also propose a new means for deriving an asymptotic estimate for the expected width, which is the number of nodes at the most abundant level. Crucial to our methods of proof is the uniformity achieved by the singularity analysis.