Coupling some conditioned L{é}vy trees with the Kesten tree

We consider locally compact L{\'e}vy trees conditioned to be large, with respect to different criterion: its height, its maximal ''size'' vertex and its total ''mass''. In the critical case, we provide a coupling with a truncated Kesten tree which then allows to directly prove the local convergence in distribution of the conditioned L{\'e}vy tree to be large towards the Kesten tree. We also consider the sub-critical and super-critical cases. In the former case the results can be partial, due to a possible condensation phenomenon which is outside the mathematical framework used in this paper.