A depth-first analog to the Lipschitz based Piyavskii-Shubert global maximization algorithm is presented. To within any given tolerance, the algorithm is shown to return a global maximum and maximizer for a uni-variate Lipschitz continuous function. This result is extended to a broader class of uni-variate functions. Empirical comparisons of Piyavskii-Shubert with several variations of the new algorithm are made.