Galton-Watson Probability Contraction

We are concerned with exploring the probabilities of first order statements for Galton-Watson 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 $\mathrm{\Sigma }$, indexed by $1\le j\le m$, denote the finite set of equivalence classes arising out of this game, and $D$ the set of all probability distributions over $\mathrm{\Sigma }$. Let $x_j(c)$ denote the true probability of the class $j\in \mathrm{\Sigma }$ under $Poisson(c)$ regime, and $\vec{x}(c)$ the true probability vector over all the equivalence classes. Then we are able to define a natural recursion function $\mathrm{\Gamma }$, and a map $\Psi = \Psi_{c}: D \rightarrow D$suchthat$\vec{x}©$isafixedpointof$\Psi_{c}$,andstartingwithanydistribution$\vec{x} \in D$, we converge to this fixed point via $\mathrm{\Psi }$ because it is a contraction. We show this both for $c\le 1$ and $c>1$, though the techniques for these two ranges are quite different.