I will speak about a common work with García Barroso and González P\'erez. Favre and Jonsson described in 2004 from various viewpoints a structure of real tree on the projectified space of semivaluations centered at a smooth point of a complex surface. We describe it from another viewpoint, as the projective limit of the Eggers-Wall trees of the reduced germs of curves at the given point. The Eggers-Wall tree measures the contacts between the various Newton-Puiseux series of a given curve singularity. Therefore, our viewpoint is adapted whenever one studies the local geometry of surfaces using such series.