  We present a bijection between some quadrangular dissections of an hexagon
and unrooted binary trees, with interesting consequences for enumeration, mesh
compression and graph sampling. Our bijection yields an efficient uniform
random sampler for 3-connected planar graphs, which turns out to be determinant
for the quadratic complexity of the current best known uniform random sampler
for labelled planar graphs [{\bf Fusy, Analysis of Algorithms 2005}]. It also
provides an encoding for the set $\mathcal{P}(n)$ of $n$-edge 3-connected
planar graphs that matches the entropy bound
$\frac1n\log_2|\mathcal{P}(n)|=2+o(1)$ bits per edge (bpe). This solves a
theoretical problem recently raised in mesh compression, as these graphs
abstract the combinatorial part of meshes with spherical topology. We also
achieve the {optimal parametric rate} $\frac1n\log_2|\mathcal{P}(n,i,j)|$ bpe
for graphs of $\mathcal{P}(n)$ with $i$ vertices and $j$ faces, matching in
particular the optimal rate for triangulations. Our encoding relies on a linear
time algorithm to compute an orientation associated to the minimal Schnyder
wood of a 3-connected planar map. This algorithm is of independent interest,
and it is for instance a key ingredient in a recent straight line drawing
algorithm for 3-connected planar graphs [\bf Bonichon et al., Graph Drawing
2005].
