A tight-binding model in one dimension with hierarchical potential strength is investigated. The problem is reduced to three equivalent recursive relations: (i) for characteristic polynomials, (ii) for a renormalization-group transformation, and (iii) for traces of transfer matrices. On this basis, the nature of the energy spectrum and the character of the wave functions is elucidated. The scaling properties are also analyzed in terms of the stability of fixed points, period two-cycles, and aperiodic solutions. For short chains, resistance and transmission are calculated as functions of system length and energy in view of experimental realizations. © 1989 The American Physical Society.