  We study the natural map eta between a group of binary planar trees whose
leaves are labeled by elements of a free abelian group H and a certain group
D(H) derived from the free Lie algebra over H. Both of these groups arise in
several different topological contexts. The map eta is known to be an
isomorphism over Q, but not over Z. We determine its cokernel and attack the
conjecture that it is injective.
