In this paper we give a survey of the symbolic operator methods to do statistics on random trees. We present some examples and apply the techniques to find their asymptotic behaviour. 1 Introduction Let us consider a class E of combinatorial objects, let A be an algorithm defined over the class E, and let denote the complexity measure we are interested in. Such a class E of combinatorial objects consists on a set, usually denoted by the same name as the class, and a size measure j \Delta j E : E \Gamma! IN. The subscript E in j \Delta j E will be dropped whenever it is clear from the context. We shall denote by E n the set of objects in E of size n. To analyze the average behaviour of A on an input e 2 E n with respect to measure means to compute A (n) = EfA (e) j e 2 E n g; (1:1) where EfXg denotes the expectation of the random variable X [Knu68, VF90]. By definition of expectation, Equation (1.1) can be written as A (n) = X k k PrfA (e) = k j e 2 E n g = X e2En Prfeg ...

method in random trees and moments of high orders