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Simplicial properties of the set of planar binary trees

**Abstract**

Planar binary trees appear as main ingredient of a new homology theory related
to dialgebras, c.f. [Loday 41].
Here we investigate the simplicial properties of the set 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 whose associated chain complex is acyclic.
Our main idea consists in decomposing the set of trees into classes,
by exploiting the orientation of their leaves.
This decomposition yields a chain bicomplex whose total chain complex is that
of binary trees.
Our main theorem 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, c.f. [Frabetti 5].

**Contents**

Introduction

1 - Double simplicial structure on the set of binary trees

1.1 - Faces and degeneracies on the set of binary trees

1.2 - Classes of planar binary trees and the bicomplex of trees

2- Decomposition of the bicomplex of trees into towers

2.1 - New kind of degeneracies: grafting operators.

2.1 - Decomposition of the vertical complexes into towers.

2.1 - Drawings of vertical towers.

A - Appendix. Cardinality of the classes of planar binary trees

B - Appendix. An invariant of the towers

References