•Three heuristic algorithms for reliability estimation.•Reducing redundant samplings using breadth-first search of a grid tree.•Securing a prescribed accuracy of reliability.•Detecting large curvatures on the limit state surface.•Fast computation of the reliability index via a deleting process. A complete search of the input space is crucial for securing the accuracy of reliability estimation, but conventional search algorithm-based methods require a large number of samples to visit the entire input space. To this end, this paper presents three heuristic algorithms for reliability estimation based on breadth-first search (BFS) of a grid tree (GT), namely the reliability accuracy supervised search algorithm (RASSA), the limit state surface resolution supervised search algorithm (LSSRSSA), and the reliability index precision supervised search algorithm (RIPSSA). All the proposed algorithms are characterized by traversing the entire input space through a GT while simultaneously reducing redundant samplings through BFS, and each one has its own special advantage as follows: RASSA can guarantee a prescribed accuracy of reliability estimation; LSSRSSA is able to probe large curvatures on limit-state surfaces; and RIPSSA quickly computes the reliability index. The computational costs and limitations of the proposed algorithms are analyzed. In addition, the accuracy, efficiency, and practicality of the proposed algorithm are validated through comparisons with other methods and an engineering application.