Cutting down random trees

Source abstract

Let T n denote a tree with n (≧ 2) labelled points: we assume T n 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 T n , then T n falls into two subtrees one of which, say T k , contains the root x. If k ≧ 2 we can remove some edge of T k and obtain an even smaller subtree of T n that contains x . If we repeat this process we will eventually obtain the subtree consisting of x itself. Let λ = λ( T n ) denote the number of edges removed from T n before the root x is isolated. Our main object here is to determine the expected value μ( n ) and variance σ 2 ( n ) of λ( T n ) under the assumptions (1) T n is chosen at random from the set of n n−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 ) ~ (½π n )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.