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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. [ABSTRACT FROM AUTHOR]

We give an explicit construction of the increasing tree-valued process introduced by Abraham and Delmas using a random point process of trees and a grafting procedure. This random point process will be used in companion papers to study record processes on Lévy trees. We use the Poissonian structure of the jumps of the increasing tree-valued process to describe its behavior at the first time the tree grows higher than a given height, using a spinal decomposition of the tree, similar to the classical Bismut and Williams decompositions. We also give the joint distribution of this exit time and the ascension time which corresponds to the first infinite jump of the tree-valued process. [ABSTRACT FROM AUTHOR]