Additive tree functionals allow to represent the cost of many divide-and-conquer algorithms. We give an invariance principle for such tree functionals for the Catalan model (random tree uniformly distributed among the full binary ordered trees with given number of nodes) and for simply generated trees (including random tree uniformly distributed among the ordered trees with given number of nodes). In the Catalan model, this relies on the natural embedding of binary trees into the Brownian excursion and then on elementary L 2 computations. We recover results first given by Fill and Kapur (2004) and then by Fill and Janson (2009). In the simply generated case, we use convergence of conditioned Galton-Watson towards stable Lévy trees, which provides less precise results but leads us to conjecture a different phase transition value between "global" and "local" regime. We also recover results first given by Janson (2003 and2016) in the quadratic case and give a generalization to the stable case.

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables. This result is a first step for giving a nonparametric test for identifying the degree function of a large random graph. Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.

We give asymptotics for the cumulative distribution function (CDF) for degrees of large dense random graphs sampled from a graphon. The proof is based on precise asymptotics for binomial random variables.Replacing the indicator function in the empirical CDF by a smoother function, we get general asymptotic results for functionals of homomorphism densities for partially labeled graphs with smoother functions. This general setting allows to recover recent results on asymptotics for homomorphism densities of sampled graphon.

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