We present two new motion planning algorithms, based on the rapidly exploring random tree (RRT) family of algorithms. These algorithms represent the first work in the direction of derandomizing RRTs; this is a very challenging problem due to the way randomization is used in RRTs. In RRTs, randomization is used to create Voronoi bias, which causes the search trees to rapidly explore the state space. Our algorithms take steps to increase the Voronoi bias and to retain this property without the use of randomization. Studying these and related algorithms would improve our understanding of how efficient exploration can be accomplished, and would hopefully lead to improved planners. We give experimental results that illustrate how the new algorithms explore the state space and how they compare with existing RRT algorithms.