  We show that the unit area Liouville quantum gravity sphere can be
constructed in two equivalent ways. The first, which was introduced by the
authors and Duplantier, uses a Bessel excursion measure to produce a Gaussian
free field variant on the cylinder. The second uses a correlated Brownian loop
and a "mating of trees" to produce a Liouville quantum gravity sphere decorated
by a space-filling path.
  In the special case that $\gamma=\sqrt{8/3}$, we present a third equivalent
construction, which uses the excursion measure of a $3/2$-stable L\'evy process
(with only upward jumps) to produce a pair of trees of quantum disks that can
be mated to produce a sphere decorated by SLE$_6$. This construction is
relevant to a program for showing that the $\gamma=\sqrt{8/3}$ Liouville
quantum gravity sphere is equivalent to the Brownian map.
