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In recent years, the use of random planar maps as discretized random surfaces has received a considerable attention in the physicists community. It is believed that the large-scale properties, or the scaling limit of these objects should not depend on the local properties of these maps, a phenomenon called universality.
By using a bijection due to Bouttier-di Francesco-Guitter between certain classes of planar maps and certain decorated trees, we give instances of such universality
phenomenons when the random maps follow a Boltzmann distribution where each face with degree $2i$ receives a nonnegative weight $q(i)$. For example, we show that under
certain regularity hypothesis for the weight sequence, the radius of the random map conditioned to have $n$ faces scales as $n^{1/4}$, as predicted by physicists and shown in the case of quadrangulations by Chassaing and Schaeffer. Our main tool is a new invariance principle for multitype Galton-Watson trees and discrete snakes.