  We study fine properties of the so-called stable trees, which are the scaling
limits of critical Galton-Watson trees conditioned to be large. In particular
we derive the exact Hausdorff measure function for Aldous' continuum random
tree and for its level sets. It follows that both the uniform measure on the
tree and the local time measure on a level set coincide with certain Hausdorff
measures. Slightly less precise results are obtained for the Hausdorff measure
of general stable trees.
