Simply generated trees, B-Series and wigner processes

We consider simply generated trees, like rooted plane trees, and consider the problem of computing generating functions of so-called bare functionals, like the tree factorial, using B-scries from Butcher's theory. We exhibit a special class of functionals from probability theory: the associated generating functions can be seen as limiting traces of product of semi-circular elements. (C) 2004 Wiley Periodicals, Inc.