  In this paper we study the asymptotic behavior of Brownian motion in both
comb-shaped planar domains, and comb-shaped graphs. We show convergence to a
limiting process when both the spacing between the teeth \emph{and} the width
of the teeth vanish at the same rate. The limiting process exhibits an
anomalous diffusive behavior and can be described as a Brownian motion
time-changed by the local time of an independent sticky Brownian motion. In the
two dimensional setting the main technical step is an oscillation estimate for
a Neumann problem, which we prove here using a probabilistic argument. In the
one dimensional setting we provide both a direct SDE proof, and a proof using
the trapped Brownian motion framework in Ben Arous \etal (Ann.\ Probab.\ '15).
