We prove that the multiplicity of a fixed eigenvalue alpha in a random recursive tree on n vertices satisfies a central limit theorem with mean and variance asymptotically equal to mu alpha n and sigma alpha 2n respectively. It is also shown that mu alpha and sigma alpha 2 are positive for every totally real algebraic integer. The proofs are based on a general result on additive tree functionals due to Holmgren and Janson. In the case of the eigenvalue 0, the constants mu 0 and sigma 02 can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.