Galton-Watson probability contraction

We are concerned with exploring the probabilities of first order statements for GaltonWatson trees with Poisson (c) offspring distribution. Fixing a positive integer k, we exploit the k-move Ehrenfeucht game on rooted trees for this purpose. Let Sigma, indexed by 1 <= j <= m, denote the finite set of equivalence classes arising out of this game, and D the set of all probability distributions over Sigma. Let x (j) (c) denote the true probability of the class j epsilon Sigma under Poisson (c) regime, and (x) over bar (c) the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function, and a map psi = psi(c) : D -> D such that (x) over bar (c) is a fixed point of psi(c), and starting with any distribution (x) over bar epsilon D, we converge to this fixed point via psi because it is a contraction. We show this both for c <= 1 and c > 1, though the techniques for these two ranges are quite different.