  We give a counterexample to a conjecture of Poon [Poo06] that any orthogonal
tree in two dimensions can always be flattened by a continuous motion that
preserves edge lengths and avoids self-intersection. We show our example is
locked by extending results on strongly locked self-touching linkages due to
Connelly, Demaine and Rote [CDR02] to allow zero-length edges as defined in
[ADG07], which may be of independent interest. Our results also yield a locked
tree with only eleven edges, which is the smallest known example of a locked
tree.
