Cost functionals for large (uniform and simply generated) random trees

Abstract

Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered trees with given number of nodes) and for simply generated trees (including random tree uniformly distributed among the ordered trees with given number of nodes). In the Catalan model, this relies on the natural embedding of binary trees into the Brownian excursion and then on elementary L 2 computations. We recover results first given by Fill and Kapur (2004) and then by Fill and Janson (2009). In the simply generated case, we use convergence of conditioned Galton-Watson towards stable Lévy trees, which provides less precise results but leads us to conjecture a different phase transition value between "global" and "local" regime. We also recover results first given by Janson (2003 and2016) in the quadratic case and give a generalization to the stable case.

Protection numbers in simply generated trees and Pólya trees

“…This is the class of complete binary trees with only internal vertices contributing to the size. The generating function is then dened by the functional equation B(z) = 1 + zB(z) 2 with B(z) = C(z)/z where C(z) is the function displayed in (5). Though this class does not strictly fall into the simply generated framework, the functional equation is of the form B(z)−1 = zφ(B(z)−1), which reects the fact that incomplete binary trees with all nodes counted are in bijection to complete binary trees with only internal vertices counted.…”

“…For simply generated trees a general theory of asymptotics of additive cost functionals was developed recently in [5], but this theory, which is based on embeddings into Brownian excursion and weak limit theorems, does not cover functionals in the local regime (i.e., functional with small toll functions), such as the number of protected nodes. Devroye and Janson [7] presented a unied approach to obtaining the number of k-protected nodes in various classes or random trees by putting them in the general context of fringe subtrees introduced by Aldous in [1].…”

Limiting Behavior Of Additive Functionals On The Stable Tree

“…height) of the subtree of T above level r containing x. Such functionals arise as scaling limits of additive functionals of the size and height on conditioned Bienaymé-Galton-Watson trees, see Delmas, Dhersin and Sciauveau [7] or Abraham, Delmas and Nassif [2] where it is shown that Z α,β < ∞ a.s. if (and only if) γα + (γ − 1)(β + 1) > 0, see Corollary 6.10 therein. In the present paper, we only consider α, β 0 which guarantees the finiteness of Z α,β .…”

Protection numbers in simply generated trees and Pólya trees

“…For simply generated trees a general theory of asymptotics of certain functional was developed recently in [6], but this theory does not cover local functionals as the number of protected nodes. Devroye and Janson [8] presented a unified approach to obtaining the number of k-protected nodes in various classes or random trees by putting them in the general context of fringe subtrees introduced by Aldous in [2].…”

“…(6). The approximate values for the limits of mean and variance of the protection number of a random vertex in different classes of simply generated trees.…”