We consider random binary trees that appear as the output of certain standard algorithms for sorting and searching if the input is random. We introduce the subtree size metric on search trees and show that the resulting metric spaces converge with probability 1. This is then used to obtain almost sure convergence for various tree functionals, together with representations of the respective limit random variables as functions of the limit tree.
"Search trees: Metric aspects and strong limit theorems." Ann. Appl. Probab. 24 (3) 1269 - 1297, June 2014. https://doi.org/10.1214/13-AAP948