The structure of Galton-Watson trees conditioned to be of a given size is well-understood. We provide yet another embedding theorem that permits us to obtain asymptotic probabilities of events that are determined by what happens near the root of these trees. As an example, we derive the probability that a Galton-Watson tree is cut when each node is independently removed with probability p, where by cutting a tree we mean that every path from root to leaf must have at least one removed node.
"A Note on the Probability of Cutting a Galton-Watson Tree." Electron. J. Probab. 16 2001 - 2019, 2011. https://doi.org/10.1214/EJP.v16-952