This talk will focus on the rooted Galton-Watson (GW) tree. The offspring distribution we consider is Poisson(\lambda), but our results extend to more general distributions. First order properties on rooted trees capture the local, finite structures inside the tree. We analyze the probabilities of first order properties under the GW measure, and obtain these probabilities as fixed points of contracting distributional maps. Moreover, we come up with nice functions that express these probabilities conditioned on survival of the GW tree. This is joint work with Joel Spencer.