  This is the third of three papers that refine and extend portions of our
earlier preprint, "The depth of a knot tunnel." Together, they rework the
entire preprint.
  In this paper, we use the theory of tunnel number 1 knots that we introduced
in "The tree of knot tunnels" to strengthen the Tunnel Leveling Theorem of H.
Goda, M. Scharlemann, and A. Thompson. This yields considerable information
about bridge numbers of tunnel number 1 knots. In particular, we calculate the
minimum bridge number of a knot as a function of the maximum depth invariant d
of its tunnels. The growth of this value is on the order of (1+\sqrt{2})^d. We
also find the maximum bridge number as a function of the number of cabling
constructions needed to produce the tunnel, showing in particular that the
maximum bridge number of a knot produced by n cabling constructions is the
(n+2)nd Fibonacci number. Finally, we examine the special case of the "middle"
tunnels of torus knots.
