Arithmetree

We construct an addition and a multiplication on the set of planar binary trees, closely related to addition and multiplication on the integers. This gives rise to a new kind of (noncommutative) arithmetic theory. The price to pay for this generalization is that, first, the addition is not commutative, second, the multiplication is distributive with the addition only on the left. This algebraic structure is the “exponent part” of the free dendriform algebra on one generator, a notion related to several other types of algebras. In the second part we extend this theory to all the planar trees. Then it is related to the free dendriform trialgebra as constructed in [J.-L. Loday, M.O. Ronco, C.R. Acad. Sci. Paris Ser. I 333 (2001) 81–86].