Local limits of conditioned Galton-Watson trees: The infinite spine case

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 limits of conditioned Galton- Watson trees. We then apply this condition to get new results in the critical case (with a general offspring distribution) and in the sub-critical cases (with a generic offspring distribution) 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.