  The present lecture notes give an introduction to works of Garc\'{i}a
Barroso, Gonz\'alez P\'erez, Ruggiero and the author. The starting point of
those works is a theorem of P\l oski, stating that one defines an ultrametric
on the set of branches drawn on a smooth surface singularity by associating to
any pair of distinct branches the quotient of the product of their
multiplicities by their intersection number. We show how to construct
ultrametrics on certain sets of branches drawn on any normal surface
singularity from their mutual intersection numbers and how to interpret the
associated rooted trees in terms of the dual graphs of adapted embedded
resolutions. The text begins by recalling basic properties of intersection
numbers and multiplicities on smooth surface singularities and the relation
between ultrametrics on finite sets and rooted trees. On arbitrary normal
surface singularities one has to use Mumford's definition of intersection
numbers of curve singularities drawn on them, which is also recalled.
