It has recently been shown that any simple (i.e. nonintersecting) polygonal chain in the plane can be reconfigured to lie on a straight line, and any simple polygon can be reconfigured to be convex. This result cannot be extended to tree linkages: we show that there are trees with two simple configurations that are not connected by a motion that preserves simplicity throughout the motion. Indeed, we prove that an N-link tree can have 2Ω(N) equivalence classes of configurations.
Biedl, Therese; Demaine, Erik D.; Demaine, Martin L.; Lazard, Sylvain; Lubiw, Anna; O'Rourke, Joseph; Robbins, Steve; Streinu, Ileana; Toussaint, Godfried; and Whitesides, Sue, "On Reconfiguring Tree Linkages: Trees Can Lock" (2000). Faculty Publications. Paper 72.