Global regime for general additive functionals of conditioned Bienaymé-Galton-Watson trees

We give an invariance principle for very general additive functionals of conditioned Bienaymé-Galton-Watson trees in the global regime when the offspring distribution lies in the domain of attraction of a stable distribution, the limit being an additive functional of a stable Lévy tree. This includes the case when the offspring distribution has finite variance (the Lévy tree being then the Brownian tree). We also describe, using an integral test, a phase transition for toll functions depending on the size and height.