  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.
