  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 Loday-Ronco's paper "Une
dualite' entre simplexes standards et polytopes de Stasheff".
