  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 $\Sigma$, indexed by $1 \leq j \leq m$, 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 $j \in \Sigma$ 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 $\Gamma$, and a map $\Psi =
\Psi_{c}: D \rightarrow D$ such that $\vec{x}(c)$ is a fixed point of
$\Psi_{c}$, and starting with any distribution $\vec{x} \in D$, we converge to
this fixed point via $\Psi$ because it is a contraction. We show this both for
$c \leq 1$ and $c > 1$, though the techniques for these two ranges are quite
different.

Probability