The possibility of fractal benzenoids built upon a (deterministically) regular self-similar structure are proposed. Of various possibilities considered a simpler class of trigonal structures is identified. Possible directed synthetic schemes are described. Further, Kekul structure counts are made as well as conjugated-circuits estimates of resonance energies. Results, for the general trigonal class as well as for particular cases are given. 1991, Taylor & Francis Group, LLC. All rights reserved.