On the distribution of eigenvalues of increasing trees

We prove that the multiplicity of a fixed eigenvalue α in a random recursive tree on n vertices satisfies a central limit theorem with mean and variance asymptotically equal to μ_αn and σ_α²n respectively. It is also shown that μ_α and σ_α² 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 μ₀ and σ₀² can be determined explicitly by means of generating functions. Analogous results are also obtained for Laplacian eigenvalues and binary increasing trees.