Labeled binary planar trees and quasi-Lie algebras

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.