Left and right length of paths in binary trees or on a question of Knuth

We consider extended binary trees and study the common right and left depth of leaf j, where the leaves are labelled from left to right by 0, 1, . . . , n, and the common right and left external pathlength of binary trees of size n. Under the random tree model, i.e., the Catalan model, we characterize the common limiting distribution of the suitably scaled left depth and the difference between the right and the left depth of leaf j in a random size-n binary tree when j ∼ ρn with 0 < ρ < 1, as well as the common limiting distribution of the suitably scaled left external pathlength and the difference between the right and the left external pathlength of a random size-n binary tree.