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We consider the Brownian tree introduced by Aldous and the associated Q-process which consists in an infinite spine on which are grafted independent Brownian trees. We present a reversal procedure on these trees that consists in looking at the tree downward from its top: the branching points becoming leaves and leaves becoming branching points. We prove that the distribution of this tree is invariant under this reversal procedure, which provides a better understanding of previous results from Bi and Delmas (2016). [ABSTRACT FROM AUTHOR]

Representation of coalescent processes using pruning of trees has been used by Goldschmidt and Martin for the Bolthausen-Sznitman coalescent and by Abraham and Delmas for the β(3/2, 1/2)-coalescent. By considering a pruning procedure on stable Galton-Watson tree with n labeled leaves, we give a representation of the discrete β(1+α, 1-α)-coalescent, with α ∈ [1/2, 1) starting from the trivial partition of the n first integers. The construction can also be made directly on the stable continuum Lévy tree, with parameter 1/α, simultaneously for all n. This representation allows to use results on the asymptotic number of coalescence events to get the asymptotic number of cuts in stable Galton-Watson tree (with infinite variance for the offspring distribution) needed to isolate the root. Using convergence of the stable Galton-Watson tree conditioned to have infinitely many leaves, one can get the asymptotic distribution of blocks in the last coalescence event in the β(1 + α, 1 - α)-coalescent. [ABSTRACT FROM AUTHOR]