  We study a configuration model on bipartite planar maps in which, given $n$
even integers, one samples a planar map with $n$ faces uniformly at random with
these face degrees. We prove that when suitably rescaled, such maps always
admit nontrivial subsequential limits as $n \to \infty$ in the
Gromov-Hausdorff-Prokhorov topology. Further, we show that they converge in
distribution towards the celebrated Brownian sphere, and more generally a
Brownian disk for maps with a boundary, if and only if there is no inner face
with a macroscopic degree, or, if the perimeter is too big, the maps degenerate
and converge to the Brownian tree. By first sampling the degrees at random with
an appropriate distribution, this model recovers that of size-conditioned
Boltzmann maps associated with critical weights in the domain of attraction of
a stable law with index $\alpha\in [1,2]$. The Brownian tree and disks then
appear respectively in the case $\alpha=1$ and $\alpha=2$, whereas in the case
$\alpha \in (1,2)$ our results partially recover previous known ones. Our
proofs rely on known bijections with labelled plane trees, which are similarly
sampled uniformly at random given $n$ outdegrees. Along the way, we obtain some
results on the geometry of such trees, such as a convergence to the Brownian
tree but only in the weaker sense of subtrees spanned by random vertices, which
are of independent interest.
