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We study the maximal degree of (sub)critical L{\'e}vy trees which arise asthe scaling limits of Bienaym{\'e}-Galton-Watson trees. We determine thegenealogical structure of large nodes and establish a Poissonian decompositionof the tree along those nodes. Furthermore, we make sense of the distributionof the L{\'e}vy tree conditioned to have a fixed maximal degree. In the casewhere the L{\'e}vy measure is diffuse, we show that the maximal degree isrealized by a unique node whose height is exponentially distributed and we alsoprove that the conditioned L{\'e}vy tree can be obtained by grafting a L{\'e}vyforest on an independent size-biased L{\'e}vy tree with a degree constraint ata uniformly chosen leaf. Finally, we show that the L{\'e}vy tree conditioned onhaving large maximal degree converges locally to an immortal tree (which is thecontinuous analogue of the Kesten tree) in the critical case and to acondensation tree in the subcritical case. Our results are formulated in termsof the exploration process which allows to drop the Grey condition.

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