By application of a general procedure devised by Martin, we generated, up to N=14, the number of self-avoiding open chains, and determined their mean-square end-to-end distance, their radius of gyration, the number of returns to the origin, and its corresponding mean-square end-to-end distances. The self-avoiding chain results were in excellent agreement with Monte Carlo calculations, and the mean-square radius of gyration of ring systems agreed with our previous Monte Carlo estimates. The number of returns to the origin was used to calculate the order of a phase transition for a tetrahedral model of the helix-to-randomcoil system. The higher-order transition found is the same as that previously obtained by Fisher for other three-dimensional model systems.