  Let $K$ be a compact subset of $\bar{\bold C} ={\bold R}^2$ and let $K^c$
denote its complement. We say $K\in HR$, $K$ is holomorphically removable, if
whenever $F:\bar{\bold C} \to\bar{\bold C}$ is a homeomorphism and $F$ is
holomorphic off $K$, then $F$ is a M\"obius transformation. By composing with a
M\"obius transform, we may assume $F(\infty )=\infty$. The contribution of this
paper is to show that a large class of sets are $HR$. Our motivation for these
results is that these sets occur naturally (e.g. as certain Julia sets) in
dynamical systems, and the property of being $HR$ plays an important role in
the Douady-Hubbard description of their structure.
