Conditioning (sub)critical L{\'e}vy trees by their maximal degree: Decomposition and local limit

We study the maximal degree of (sub)critical L{\'e}vy trees which arise as the scaling limits of Bienaym{\'e}-Galton-Watson trees. We determine the genealogical structure of large nodes and establish a Poissonian decomposition of the tree along those nodes. Furthermore, we make sense of the distribution of the L{\'e}vy tree conditioned to have a fixed maximal degree. In the case where the L{\'e}vy measure is diffuse, we show that the maximal degree is realized by a unique node whose height is exponentially distributed and we also prove that the conditioned L{\'e}vy tree can be obtained by grafting a L{\'e}vy forest on an independent size-biased L{\'e}vy tree with a degree constraint at a uniformly chosen leaf. Finally, we show that the L{\'e}vy tree conditioned on having large maximal degree converges locally to an immortal tree (which is the continuous analogue of the Kesten tree) in the critical case and to a condensation tree in the subcritical case. Our results are formulated in terms of the exploration process which allows to drop the Grey condition.