Mathematisches Forschungsinstitut Oberwolfach Report No . 41 / 2004 Mini-Workshop : Probability Theory on Trees and Analysis of Algorithms Organised

Random records and cuttings in trees Svante Janson We consider random cutting down of rooted trees, defined as follows [6]. If T is a rooted tree with number of vertices |T | ≥ 2, we make a random cut by choosing one edge at random. Delete this edge so that the tree separates into two parts, and keep only the part containing the root. Continue recursively until only the root is left. We let X(T ) denote the (random) number of cuts that are performed until the tree is gone. The same random variable appears when we consider records in a tree. Let each edge e have a random value λe attached to it, and assume that these values are i.i. d. with a continuous distribution. Say that a value λe is a record if it is the largest value in the path from the root to e. Then the number of records is again given by X(T ), as is easily seen. There are also vertex versions of cuttings and records. For cuttings, choose a vertex at random and destroy it together with all its descendants. Continue until the root is chosen and thus the whole tree is destroyed. For records, we assign i.i. d. values λv to the vertices, and define a record as above. Again, there is an equivalence between cuttings and records. The edge and vertex versions are closely related, and the results are essentially the same. These random variables can be studied both for deterministic trees and for random trees. If the tree is a path, we have the classical record problem studied by Rényi, [9]. Our main results are for the case when the tree Tn itself is random, more precisely a random conditioned Galton–Watson tree (also known as simply generated tree) with n vertices. (It is well-known that examples include random labelled trees and random binary trees.) Since both the records (or cutting) and the tree now are random, X(Tn) can be regarded in (at least) two ways. First, we can regardX(Tn) as a random variable, obtained by picking a random tree Tn and then a random cutting of it. This point of view has been taken by Meir and Moon [6] (mean and variance for Cayley trees), Chassaing and Marchand [3] (asymptotic distribution for Cayley trees), Panholzer [7, 8] (asymptotic distribution for some special families of simply generated trees, and for non-crossing trees). We extend these results to all conditioned Galton–Watson trees. (All unspecified limits are as n→ ∞.) Theorem 1. Let Tn be a conditioned Galton–Watson tree of size n, defined by an offspring distribution ξ with mean E ξ = 1 and finite variance σ > 0. Then, X(Tn) σn1/2 d −→ Z, 2138 Oberwolfach Report 41/2004 where Z has a Rayleigh distribution with density xe−x 2/2, x > 0. Moreover, if E ξ <∞ for every m > 0, then all moments converge and thus, for every r > 0, EX(Tn) r ∼ σn EZ = 2σΓ ( r 2 + 1 ) n. The other point of view is to study X(Tn) as a random variable conditioned on Tn. In other words, we consider the random procedure in two steps: First we choose a random tree T = Tn. Then we keep this tree fixed and consider random cuttings of it; this gives a random variable X(T ) with a distribution that depends on T . Normalizing as in the theorem above, we consider the distribution of σ−1n−1/2X(Tn) given Tn; this is thus a random probability distribution. We then can show that this random probability distribution converges in distribution to a random probability distribution (that does not depend on ξ); this random distribution has moments that can be expressed as functionals of a Brownian excursion. The proofs are based on Aldous’ theory of the continuum random tree [1, 2]. Finally, we study the case when the tree is a (deterministic) complete binary tree of size n. In this case, both the methods and results are different. There is now a periodicity in the result. This is not surprising for complete binary trees, but it is a bit surprising that the periodicity is in the fractional part {lgn−lg lgn}. Theorem 2. Suppose that n→ ∞ such that {lgn− lg lg n} → γ ∈ [0, 1]. Then ( X(Tn) − n lg n − n lg lg n lg n ) / n lg n d −→ −Wγ where Wγ has an infinitely divisible distribution with characteristic function E eγ = exp (