  According to an analogy to quasi-Fuchsian groups, we investigate topological
and combinatorial structures of Lyubich and Minsky's affine and hyperbolic
3-laminations associated with the hyperbolic and parabolic quadratic maps.
  We begin by showing that hyperbolic rational maps in the same hyperbolic
component have quasi-isometrically the same 3-laminations. This gives a good
reason to regard the main cardioid of the Mandelbrot set as an analogue of the
Bers slices in the quasi-Fuchsian space. Then we describe the topological and
combinatorial changes of laminations associated with hyperbolic-to-parabolic
degenerations (and parabolic-to-hyperbolic bifurcations) of quadratic maps. For
example, the differences between the structures of the quotient 3-laminations
of Douady's rabbit, the Cauliflower, and $z \mapsto z^2$ are described.
  The descriptions employ a new method of tessellation inside the filled Julia
set introduced in Part I that works like external rays outside the Julia set.
