Expressions for the multi-dimensional densities of Brownian excursion local time are derived by two different methods: A direct method based on Kac’s formula for Brownian functionals and an indirect one based on a limit theorem for Galton-Watson trees.

We propose martingale central limit theorems as an appropriate tool to prove asymptotic normality of the costs of certain recursive algorithms which are subjected to random input data. The recursive algorithms that we have in mind are such that if input data of size N produce random costs LN, then LN D = Ln + ¯LN−n + RN for N ≥ n0 ≥ 2, where n follows a certain distribution PN on the integers {0,...,N} and Lk D = ¯Lk for k ≥ 0. Ln, LN−n and RN are independent, conditional on n, and RN are random variables, which may also depend on n, corresponding to the cost of splitting the input data of size N (into subsets of size n and N − n) and combining the results of the recursive calls to yield the overall result. We construct a martingale difference array with rows converging to ZN: = LN−IE LN √. Under certain Var LN compatibility assumptions on the sequence (PN)N≥0 we show that a pair of sufficient conditions (of Lyapunov type) for ZN D → N (0,1) can be restated as a pair of conditions regarding asymptotic relations between three sequences. All these sequences satisfy the same type of linear equation, that is also the defining equation for the sequence (IE LN)N≥0. In the case that the PN are binomial distributions with the same parameter p, and for deterministic RN, we demonstrate the power of this approach. We derive very general sufficient conditions in terms of the sequence (RN)N≥0 (and for the scale RN = N α a characterization of those α) leading to asymptotic normality of ZN.

The purpose of this article is to present explicit and asymptotic methods to count various kinds of trees. In all cases the use of generating functions is essential. Explicit formulae are derived with help of Lagrange's inversion formula. On the other hand singularity analysis of generating functions leads to asymptotic formulas.

Abstract. Let Tn denote the set of unrooted labeled trees of size n and let M be a particular (finite, unlabeled) tree. Assuming that every tree of Tn is equally likely, it is shown that the limiting distribution as n goes to infinity of the number of occurrences of M as an induced subtree is asymptotically normal with mean value and variance asymptotically equivalent to µn and σ 2 n, respectively, where the constants µ> 0 and σ ≥ 0 are computable. 1.

It is proved that the moments of the width of Galton-Watson trees of size n and with ospring variance are asymptotically given by ( n) mp where mp are the moments of the maximum of the local time of a standard scaled Brownian excursion. This is done by combining a weak limit theorem and a tightness estimate. The method is quite general and we state some further applications. 1.

Abstract. We consider the number of nodes in the levels of unlabeled rooted random trees and show that the stochastic process given by the properly scaled level sizes weakly converges to the local time of a standard Brownian excursion. Furthermore we compute the average and the distribution of the height of such trees. These results extend existing results for conditioned Galton-Watson trees and forests to the case of unlabeled rooted trees and show that they behave in this respect essentially like a conditioned Galton-Watson process. 1.

Abstract. We study the asymptotic behaviour of the number Nk,n of nodes of given degree k in unlabeled random trees, when the tree size n and the node degree k both tend to infinity. It is shown that Nk,n is asymptotically normal if ENk,n → ∞ and asymptotically Poisson distributed if ENk,n → C> 0. If ENk,n → 0, then the distribution degenerates. The same holds for rooted, unlabeled trees and forests. 1.

Let Tn denote the set of unrooted unlabeled trees of size n and let M be a particular (finite) tree. Assuming that every tree of Tn is equally likely, it is shown that the number of occurrences Xn of M as an induced sub-tree satisfies E Xn ∼ µn and Var Xn ∼ σ 2 n for some (computable) constants µ> 0 and σ ≥ 0. Furthermore, if σ> 0 then (Xn − E Xn) / √ Var Xn converges to a limiting distribution with density (A + Bt 2)e −Ct2 for some constants A, B, C. However, in all cases in which we were able to calculate these constants, we obtained B = 0 and thus a normal distribution. Further, if we consider planted or rooted trees instead of Tn then the limiting distribution is always normal. Similar results can be proved for planar, labeled and simply generated trees.

We consider the number of nodes in the levels of unlabeled rooted random trees and show that the joint distribution of several level sizes (where the level number is scaled by √ n) weakly converges to the distribution of the local time of a Brownian excursion evaluated at the times corresponding to the level numbers. This extends existing results for

Abstract. We investigate the profile of random Pólya trees of size n when only nodes of degree d are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process converges weakly to a Brownian excursion local time. Moreover, we investigate the joint distribution of the number of nodes of degree d1 and d2 in the levels of the tree. 1.