  Let $f$ be a real- or circle-valued Morse function on a compact surface M
having exactly $n>0$ critical points. Denote by $O$ the orbit of $f$ with
respect to the right action of the group of diffeomorphisms of $M$. We show
that the connected components of $O$ have the homotopy type of a
finite-dimensional CW-complex. Actually, these connected components are
homotopy equivalent to a certain covering space of the $n$-th configuration
space of the interior of $M$. As a consequence we obtain that the fundamental
group of $O$ is a subgroup of the $n$-th braid group of $M$.
