Cutting down random trees

Abstract

Let Tn denote a tree with n(≧ 2) labelled points: we assume Tn is rooted at a given point x, say the point labelled 1 (see [3] for definitions not given here). If we remove some edge e of Tn, then Tn falls into two subtrees one of which, say Tk, contains the root x. If k ≧ 2 we can remove some edge of Tk and obtain an even smaller subtree of Tn that contains x. If we repeat this process we will eventually obtain the subtree consisting of x itself. Let λ = λ(Tn) denote the number of edges removed from Tn before the root x is isolated. Our main object here is to determine the expected value μ(n) and variance σ2(n) of λ(Tn) under the assumptions (1) Tn is chosen at random from the set of nn−2 trees with n labelled points that are rooted at point x, and (2) at each stage the edge removed is chosen at random from the edges of the remaining subree containing x. It follows from our results that μ(n) ~ (½πn)½ and (2−½π)n ~ (2−½π)n as n tends to infinity. We also consider the corresponding problem for forests of rooted trees and for trees in which the degree of the root is specified. We are indebted to Professor Alistair Lachlan for suggesting the original problem to us.

Random Records and Cuttings in Complete Binary Trees

“…The same random variables appears when we consider random cuttings of the tree T defined as follows, see [6]. Make a random cut by choosing one vertex [edge] at random.…”

Random walks on random trees

“…We shall state these asymptotic results without proof; the details of the missing arguments are quite similar to those given, for example, in [7] and [5].…”

“…x"e-x lz dx; Jo in particular, we shall frequently use the fact that if k = oin*), then n-k E (n -k) v n~" ~ (inri)* as n and k tend to infinity (see [5]). We shall state these asymptotic results without proof; the details of the missing arguments are quite similar to those given, for example, in [7] and [5].…”

Isolating a Leaf in Rooted Trees via Random Cuttings

“…Best to our knowledge the procedure studied here and the results are new, but for the sake of completeness we collect in the following some known results for the opposite procedure, which isolates the root. Meir and Moon [13,14] considered this edge-removal procedure (= cutting-down procedure) on a rooted tree with n vertices. In papers [12,13] a random variable X n was studied.…”

“…Meir and Moon [13,14] considered this edge-removal procedure (= cutting-down procedure) on a rooted tree with n vertices. In papers [12,13] a random variable X n was studied. This variable counts the number of edges that will be removed from a randomly chosen tree of size n until the root is isolated for the two important tree families unordered labelled trees (= Cayley trees) and recursive trees.…”