  We propose a new way to condition random trees, that is, condition random
trees to have large maximal out-degree. Under this new conditioning, we show
that conditioned critical Galton-Watson trees converge locally to size-biased
trees with a unique infinite spine. For the sub-critical case, we obtain local
convergence to size-biased trees with a unique infinite node. We also study
tail of the maximal out-degree of sub-critical Galton-Watson trees, which is
essential for the proof of the local convergence.
