The autonomous navigation of vehicles typically combines two kinds of methods: a path is first planned, and then the robot is driven by a local obstacle-avoidance controller. The present work, which focuses on path planning, proposes an extension to the well-known rapidly-exploring random tree (RRT) algorithm to allow its integration with a trajectory parameter-space (TP-space) as an efficient method to detect collision-free, kinematically-feasible paths for arbitrarily-shaped vehicles. In contrast to original RRT, this proposal generates navigation trees, with poses as nodes, whose edges are all kinematically-feasible paths, suitable to being accurately followed by vehicles driven by pure reactive algorithms. Initial experiments demonstrate the suitability of the method with an Ackermann-steering vehicle model whose severe kinematic constraints cannot be obviated. An important result that sets this work apart from previous research is the finding that employing several families of potential trajectories to expand the tree, which can be done efficiently under the TP-space formalism, improves the optimality of the planned trajectories. A reference C++ implementation has been released as open-source.