The lattice is a special rooted Cayley tree, generated by N successive m-fold branchings. With each point of the tree are associated a mass M and a position coordinate xi. All end points are held fixed at xi=0. The potential energy is V=½ Σi,j Kij(xi−xj)2, where Kij=K if i and j are connected neighbors and neither is an end point, Kij=αK if i and j are connected neighbors and either is a branch tip point, and Kij=0 if i and j are not connected neighbors. The allowed frequencies of vibration are obtained for two different cases: In the first case all springs are identical (α=1), and in the second case the springs connecting interior points to the branch tips are cut (α=0). In the case in which all force constants are the same, the allowed frequencies of vibration, in the limit of infinite N, are given by ω(r) = (K/M) ½[m+1 − 2m½ cosrπ]½, where r is any rational number between zero and one. The fraction of all normal modes having precisely the value ω(r) is ρ[ω(r)] = (m − 1)2 / (mq − 1), where r is expressed as the ratio r=p/q of relatively prime integers p and q. The frequency spectrum is dense within the interval (m½ − 1, m½+1); and ρ[ω] is discontinuous at every ω for which it does not vanish.