  We derive a new representation of the Brownian disk in terms of a forest of
labeled trees, where labels correspond to distances from a subset of the
boundary. We then use this representation to obtain a spatial Markov property
showing that the complement of a hull centered at a boundary point of a
Brownian disk is again a Brownian disk, with a random perimeter, and is
independent of the hull conditionally on its perimeter. Our proofs rely in part
on a study of the peeling process for triangulations with a boundary, which is
of independent interest. The results of the present work will be applied to a
continuous version of the peeling process for the Brownian half-plane in a
companion paper.
