  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 $\sigma^2_{\alpha} n$
respectively. It is also shown that $\mu_{\alpha}$ and $\sigma^2_{\alpha}$ 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 $\sigma^2_0$ can be
determined explicitly by means of generating functions. Analogous results are
also obtained for Laplacian eigenvalues and binary increasing trees.
