International audience We introduce a recursive algorithm generating random trees, which we identify as skeletons of a continuous, stable tree. We deduce a representation of a fragmentation process on these trees.

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In this paper, we redesign and simplify an algorithm due to Remy et al. for the generation of rooted planar trees that satisfies a given partition of degrees. This new version is now optimal in terms of random bit complexity, up to a multiplicative constant. We then apply a natural process "simulate-guess-and-proof" to analyze the height of a random Motzkin in function of its frequency of unary nodes. When the number of unary nodes dominates, we prove some unconventional height phenomenon (i.e. outside the universal Θ( √ n) behaviour.) ⋆ Supported by ANR Magnum project BLANC 0204 (France) ⋆⋆ Supported by ANR Magnum project BLANC 0204 (France)

International audience It has been shown by Pittel and Romik that the random surface associated with a large rectangular Youngtableau converges to a deterministic limit. We study the fluctuations from this limit along the edges of the rectangle.We show that in the corner, these fluctuations are gaussian whereas, away from the corner and when the rectangle isa square, the fluctuations are given by the Tracy-Widom distribution. Our method is based on a connection with theJacobi ensemble.