Let $G$ be a Galton-Watson tree, i.e. the family tree of a Galton-Watson branching process. For $0 \leq u \leq 1$ let $G_u$ be the subtree of $G$ obtained by retaining each edge with probability $u$ and deleting other edges. We study the tree-valued Markov process $(G_u, 0 \leq u \leq 1)$ and an analogous process $(G^*_u, 0 \leq u \leq 1)$ in which $G^*_1$ is a critical or subcritical Galton-Watson tree conditioned to be infinite. Results simplify and are further developed in the special case of Poisson$(\mu)$ offspring distribution.