  Liouville Conformal Field Theory (LCFT) on the disk describes the conformal
factor of the quantum disk, which is the natural random surface in Liouville
quantum gravity with disk topology. Fateev, Zamolodchikov and Zamolodchikov
(2000) proposed an explicit expression, the so-called the FZZ formula, for the
one-point bulk structure constant for LCFT on the disk. In this paper we give a
proof of the FZZ formula in the probabilistic framework of LCFT, which
represents the first step towards rigorously solving boundary LCFT using
conformal bootstrap. In contrast to previous works, our proof is based on
conformal welding of quantum disks and the mating-of-trees theory for Liouville
quantum gravity. As a byproduct of our proof, we also obtain the exact value of
the variance for the Brownian motion in the mating-of-trees theory. Our paper
is an essential part of an ongoing program proving integrability results for
Schramm-Loewner evolutions, LCFT, and in the mating-of-trees theory.
