  We introduce an efficient way, called Newton algorithm, to study arbitrary
ideals in C[[x,y]], using a finite succession of Newton polygons. We codify
most of the data of the algorithm in a useful combinatorial object, the Newton
tree. For instance when the ideal is of finite codimension, invariants like
integral closure and Hilbert-Samuel multiplicity were already combinatorially
determined in the very special cases of monomial or non degenerate ideals,
using the Newton polygon of the ideal. With our approach, we can generalize
these results to arbitrary ideals. In particular the Rees valuations of the
ideal will correspond to the so-called dicritical vertices of the tree, and its
Hilbert-Samuel multiplicity has a nice and easily computable description in
terms of the tree.
