Computational tools for robot design require algorithms moving between several layers of abstraction including task, morphology, kinematics, mechanism shapes, and actuation. In this paper we give a linear-time algorithm mapping from kinematics to mechanism shape for tree-structured linkages. Specifically, we take as input a tree whose nodes are axes of motion (lines which joints rotate about or translate along) along with types and sizes for joints on these axes, and a radius r for a tubular bound on the link shapes. Our algorithm outputs the geometry for a kinematic tree instantiating these specifications such that the neutral configuration has no self-intersection. Output designs have linear complexity. The algorithm approach is based on understanding the mechanism design problem as a planning problem for link shapes, and arranging the joints along their axes of motion to be appropriately spaced and oriented such that feasible, non-intersecting paths exist linking them. Since link bending is restricted by its tubular radius, this is a Dubins planning problem, and to prove the correctness of our algorithm we also prove a theorem about Dubins paths: if two point-direction pairs are separated by a plane at least 2r from each, and the directions each have non-negative dot product with the plane normal, then they are connected by a radius-r CSC Dubins path with turn angles \(\le \pi \) . We implement our design algorithm in code and provide a 3D printed example of a tubular kinematic tree. The results provide an existence proof of tubular-shaped kinematic trees implementing given axes of motion, and could be used as a starting point for further optimization in an automated or algorithm-assisted robot design system.