In the course of analyzing elementary embeddings of a rank into itself, Richard Laver investigated certain finite algebraic systems now known as the Laver tables. The $n$th Laver table is the set $\{1,\ldots,2^n\}$ equipped with the unique operation $*$ satisfying $x*1 = x+1 \mod 2^n$ and the left self distributive law $x*(y*z) = (x*y)*(x*z)$. Each finite rooted ordered binary tree has a natural evaluation in each Laver table. The purpose of this talk is to demonstrate that this evaluation map can be extended to those subtrees of the complete binary tree generated by a Galton-Watson process associated to a probability $0 < p < 1$.