  Let c be a real parameter in the Mandelbrot set, and f_c(z):= z^2 + c. We
prove a formula relating the topological entropy of f_c to the Hausdorff
dimension of the set of rays landing on the real Julia set, and to the
Hausdorff dimension of the set of rays landing on the real section of the
Mandelbrot set, to the right of the given parameter c. We then generalize the
result by looking at the entropy of Hubbard trees: namely, we relate the
Hausdorff dimension of the set of external angles which land on a certain slice
of a principal vein in the Mandelbrot set to the topological entropy of the
quadratic polynomial f_c restricted to its Hubbard tree.
