  In this paper, we give an isotopy classification of 3-bridge spheres of
3-bridge arborescent links, which are not Montesinos links. To this end, we
prove a certain refinement of a theorem of J.S. Birman and H.M. Hilden on the
relation between bridge presentations of links and Heegaard splittings of
3-manifolds. In the proof of this result, we also give an answer to a question
by K. Morimoto on the classification of genus-2 Heegaard splittings of certain
graph manifolds.
