The degree profile of random Pَlya trees

We investigate the profile of random Pَlya trees of size n when only nodes of degree d are counted in each level. It is shown that, as in the case where all nodes contribute to the profile, the suitably normalized profile process converges weakly to a Brownian excursion local time. Moreover, we investigate the joint distribution of the number of nodes of degrees d 1 and d 2 on the same level of the tree.