  In the regime of Galton-Watson trees, first order logic statements are
roughly equivalent to examining the presence of specific finite subtrees. We
consider the space of all trees with Poisson offspring distribution and show
that such finite subtrees will be almost surely present when the tree is
infinite. Introducing the notion of universal trees, we show that all first
order sentences of quantifier depth $k$ depend only on local neighbourhoods of
the root of sufficiently large radius depending on $k$. We compute the
probabilities of these neighbourhoods conditioned on the tree being infinite.
We give an almost sure theory for infinite trees.
