We present a new method for producing locally non intersecting polygon meshes of naturally branching structures. The generated polygon mesh follows the object's underlying structure as close as possible, while still producing polygon meshes that can be visualized efficiently on commonly available graphic acceleration hardware. A priori knowledge of vascular branching systems is used to derive the polygon mesh generation method. Visualization of the internal liver vessel structures and naturally looking tree stems generated by Lindenmayer-systems is used as examples. The method produce visually convincing polygon meshes that might be used in clinical applications in the future.