  We study the parameter space of unicritical polynomials $f_c:z\mapsto z^d+c$.
For complex parameters, we prove that for Lebesgue almost every $c$, the map
$f_c$ is either hyperbolic or infinitely renormalizable. For real parameters,
we prove that for Lebesgue almost every $c$, the map $f_c$ is either
hyperbolic, or Collet-Eckmann, or infinitely renormalizable. These results are
based on controlling the spacing between consecutive elements in the
``principal nest'' of parapuzzle pieces.
