The Apery algorithm for a plane singularity with two branches

The equisingularity class of a plane irreducible curve is determined by the semigroup of the curve or, equivalently, by its multiplicity sequence. For a curve with two branches, the semigroup (now a subsemigroup of ${\mathbb N}^2$) still determines the equisingularity class. We introduce the ``multiplicity tree'' for the curve, which also determines the equisingularity class, and construct an algorithm to go back and forth between the semigroup and the multiplicity tree. Moreover we characterize the multiplicity trees of plane curve singularities with two branches.