Informed sampling techniques accelerate the convergence of sampling-based motion planners by biasing sampling toward regions of the state space that are most likely to yield better solutions. However, when the current solution path contains redundant or tortuous segments, the resulting informed subset may remain unnecessarily large, slowing convergence. Our prior work addressed this issue by introducing the greedy informed set, which reduces the sampling region based on the maximum heuristic cost along the current solution path. In this article, we formally characterize the behavior of the greedy informed set within Rapidly-exploring Random Tree (RRT*)-like planners and analyze how greedy sampling affects exploration and asymptotic optimality. We then present Greedy RRT* (G-RRT*), a bi-directional anytime variant of RRT* that leverages the greedy informed set to focus sampling in the most promising regions of the search space. Experiments on abstract planning benchmarks, manipulation tasks from the MotionBenchMaker dataset, and a dual-arm Barrett WAM problem demonstrate that G-RRT* rapidly finds initial solutions and converges asymptotically to optimal paths, outperforming state-of-the-art sampling-based planners.