  We study random bipartite planar maps defined by assigning nonnegative
weights to each face of a map. We prove that for certain choices of weights a
unique large face, having degree proportional to the total number of edges in
the maps, appears when the maps are large. It is furthermore shown that as the
number of edges $n$ of the planar maps goes to infinity, the profile of
distances to a marked vertex rescaled by $n^{-1/2}$ is described by a Brownian
excursion. The planar maps, with the graph metric rescaled by $n^{-1/2}$, are
then shown to converge in distribution toward Aldous' Brownian tree in the
Gromov-Hausdorff topology. In the proofs, we rely on the Bouttier-di
Francesco-Guitter bijection between maps and labeled trees and recent results
on simply generated trees where a unique vertex of a high degree appears when
the trees are large.
