Left and Right Length of Paths in Binary Trees or on a Question of Knuth

Abstract

We consider extended binary trees and study the joint right and left depth of leaf j, where the leaves are labelled from left to right by 0, 1,..., n, and the joint right and left external pathlength of binary trees of size n. Under the random tree model, i.e., the Catalan model, we characterize the joint 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 joint 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.

A Note on Naturally Embedded Ternary Trees

“…Motivated by the results of Bousquet-Mélou [3], and Panholzer [15] for naturally embedded binary trees, and the results of Schaeffer and Jacquard [9], and Del Lungo, Del Ristoro and Penaud [6] for a subclass of ternary trees, we embedd ternary trees in the plane in the following way. The root node has position zero.…”

“…The study of tree families embedded in the plane has recently received a lot of attention. Binary trees, complete binary trees, more generally simply generated tree families, and several different families of planar trees have been considered in a series of papers [4,5,13,8,3,2,11,12,7,15]. It has been shown that embedded trees are closely related to a random measure called ISE (Integrated SuperBrownian Excursion).…”

“…However, enumerative properties of deterministically embedded trees have not been intensively studied, except for binary trees, a particular subclass of ternary trees, and families of plane trees. Motivated by the results of Bousquet-Mélou [3] and Panholzer [15] for embedded binary trees, and the results of Schaeffer and Jacquard [9], and Del Lungo, Del Ristoro and Penaud [6] for a specific subclass of embedded ternary trees, we study several combinatorial properties of ternary trees embedded in the plane in a deterministic manner. We consider the following natural embedding.…”

“…Using bijections between embedded ternary trees with no label greater than zero and non-separable rooted planar maps with n + 1 edges they obtained amongst others an explicit result for the number of such trees of size n. The aim of this work is to use a generating functions approach to study several parameters in embedded ternary trees: we analyze the distribution of the different types of depths of external nodes in ternary trees, stemming from the embedding of the tree, assuming that the external nodes are enumerated from the left to the right, or in other words with respect to an inorder-traversal. Related parameters have been studied for binary trees in [16,17,15]. Moreover, we are interested in the number of embedded ternary trees of size n where all internal nodes have label smaller or equal than j, with j ∈ N, where we extend the results of [9] and [6] for the special case j = 0 to arbitrary j ≥ 0, and also in the number of embedded ternary of size n counted with respect to the number of internal nodes with label j, with j ∈ Z.…”

Generating Functions of Embedded Trees and Lattice Paths

“…Several families of embedded trees have been studied in the literature. Binary trees, complete binary trees, several different families of planar trees and more generally simply generated tree families have been considered in a series of papers [8,9,20,13,3,2,17,18,12,22,14]: it has been showed that embedded trees naturally arise in the context of map enumeration and that properties of embedded trees are closely related to a random measure called Integrated Superbrownian Excursion. Combinatorial properties of embedded ternary trees where studied using bijections between embedded ternary trees and non-separable rooted planar maps [16,10], where the authors studied a particular subclass of embedded ternary trees named skew ternary trees [16], or left ternary trees [10], which are embedded ternary trees with no node having label greater than zero.…”