Simplicial properties of the set of planar binary trees

Planar binary trees appear as the the main ingredient of a new homology theory related to dialgebras, cf.(J.-L. Loday, C.R. Acad. Sci. Paris 321 (1995), 141-146.) Here I investigate the simplicial properties of the set of these trees, which are independent of the dialgebra context though they are reflected in the dialgebra homology. The set of planar binary trees is endowed with a natural (almost) simplicial structure which gives rise to a chain complex. The main new idea consists in decomposing the set of trees into classes, by exploiting the orientation of their leaves. (This trick has subsequently found an application in quantum electrodynamics, c.f. (C. Brouder, "On the Trees of Quantum Fields," Eur. Phys. J. C12, 535-549 (2000).) This decomposition yields a chain bicomplex whose total chain complex is that of binary trees. The main theorem of the paper concerns a further decomposition of this bicomplex. Each vertical complex is the direct sum of subcomplexes which are in bijection with the planar binary trees. This decomposition is used in the computation of dialgebra homology as a derived functor, cf. (A. Frabetti, "Dialgebra (co) Homology with Coefficients," Springer L.N.M., to appear).