Asymptotic degree distribution in random recursive trees

The distributions of vertex degrees in random recursive trees and random plane recursive trees are shown to be asymptotically normal. Formulas are given for the asymptotic variances and covariances of the number of vertices with given outdegrees. We also give functional limit theorems for the evolution as vertices are added. The proofs use some old and new results about generalized Pólya urn models. We consider generalized Pólya urns with infinitely many types, but reduce them to the finite type case.