  A decoration of the Mandelbrot set $M$ is a part of $M$ cut off by two
external rays landing at some tip of a satellite copy of $M$ attached to the
main cardioid. In this paper we consider infinitely renormalizable quadratic
polynomials satisfying the decoration condition, which means that the
combinatorics of the renormalization operators involved is selected from a
finite family of decorations. For this class of maps we prove {\it a priori}
bounds. They imply local connectivity of the corresponding Julia sets and the
Mandelbrot set at the corresponding parameter values.
