  We study the local convergence of critical Galton-Watson trees and Levy trees
under various conditionings. Assuming a very general monotonicity property on
the functional of random trees, we show that random trees conditioned to have
large functional values always converge locally to immortal trees. We also
derive a very general ratio limit property for functionals of random trees
satisfying the monotonicity property. Then we move on to study the local
convergence of critical continuous-state branching processes, and prove a
similar result. Finally we give a definition of continuum condensation trees,
which should be the correct local limits for certain subcritical Levy trees
under suitable conditionings.
