On the spectra of certain rooted trees

Let T be an unweighted tree with vertex root v which is the union of two trees T₁=(V₁,E₁), T₂=(V₂,E₂) such that V₁ ∩ V₂ = {v} and T₁ and T₂ have the property that the vertices in each of their levels have equal degree. We characterize completely the eigenvalues of the adjacency matrix and of the Laplacian matrix of T. They are the eigenvalues of symmetric tridiagonal matrices whose entries are given in terms of the vertex degrees. Moreover, we give some results about the multiplicity of the eigenvalues. Applications to some particular trees are developed.