Asymptotic Properties of Protected Nodes in Random Recursive Trees

Abstract

We investigate protected nodes in random recursive trees. The exact mean of the number of such nodes is obtained by recurrence, and a linear asymptotic equivalent follows. A nonlinear recurrence for the variance shows that the variance grows linearly, too. It follows that the number of protected nodes in a random recursive tree, upon proper scaling, converges in probability to a constant.

Limit laws for functions of fringe trees for binary search trees and random recursive trees

“…By Remark 1.26, we also obtain joint normality for several . Theorem 8.6 includes several earlier results, proved by several different methods: (8.10) was shown by Mahmoud and Ward [35] for = 2 and by Bóna [6] and Devroye and Janson [12] in general; [35] also shows (8.11)-(8.12) for = 2; (8.13) was shown by Mahmoud and Ward [36] for = 2 and by Devroye and Janson [12] in general; [36] also shows for = 2 the weaker version of (8.14) that Var(Z 2,n ) = O(1/n).…”

“…We proceed to use fringe trees to study the so-called protected nodes that recently have been studied in several types of random trees, see e.g. [6,8,12,35,36] and the references there.…”

“…For the random recursive tree, µ Z,1 = 1 2 as a consequence of Theorem 8.3 (with d = 0) and µ Z,2 = 1 2 − e −1 by Mahmoud and Ward [36] and Devroye and Janson [12]; the method in [12] is based on (1.54) as here and yields (recursively) a complicated integral expression for every , but we do not know any closed form for ≥ 3.…”

“…We will use a representation of Devroye [10,11] for the binary search tree, and a well-known bijection between binary trees and recursive trees, together with different applications of Stein's method for both normal and Poisson approximation to give both new general results on the asymptotic distributions for random variables depending on fringe trees, and more direct proofs of several earlier results in the field. We give also examples of applications of these general results, for example to the number of protected nodes in the binary search tree or random recursive tree studied by Mahmoud and Ward [35,36] and to the shape functionals for the binary search tree or random recursive tree, see Section 8.…”

Protection number in plane trees

“…A node enjoys this property if its distance to any leaf is at least 2. After this pioneering paper, a large number of papers has been published [13,6,1,10,5,12].…”