Subtree Sizes in Recursive Trees and Binary Search Trees: Berry-Esseen Bounds and Poisson Approximations

We study the number of subtrees on the fringe of random recursive trees and random binary search trees whose limit law is known to be either normal or Poisson or degenerate depending on the size of the subtree. We introduce a new approach to this problem which helps us to further clarify this phenomenon. More precisely, we derive optimal Berry-Esseen bounds and local limit theorems for the normal range and prove a Poisson approximation result as the subtree size tends to infinity.