The canonical s-process theory of heavy element formation is reformulated to allow for competition between beta decay and neutron capture at various nuclei along the path. Solutions to the resulting network equations at each branch are presented that do not require the assumption of steady flow of the neutron current. An approximate expression is also developed for the complete s-process that includes the effects of all possible branches. From the observed solar system abundances and measured capture cross sections several key branching ratios are inferred. Using recently calculated temperature-dependent beta decay rates, comparisons of adjacent branches then yield an estimate for the appropriate temperature and half-life against neutron capture at selected branch points. Possible extensions of the results to observations of freshly produced s-process material in the surfaces of individual stars are also discussed.