We give a necessary and sufficient condition for the convergence in distribution of a conditioned Galton-Watson tree to Kesten's tree. This yields elementary proofs of Kesten's result as well as other known results on local limit of conditioned Galton-Watson trees. We then apply this condition to get new results, in the critical and sub-critical cases, on the limit in distribution of a Galton-Watson tree conditioned on having a large number of individuals with out-degree in a given set.
"Local limits of conditioned Galton-Watson trees: the infinite spine case." Electron. J. Probab. 19 1 - 19, 2014. https://doi.org/10.1214/EJP.v19-2747