Let
L
L
be a fixed branch – that is, an irreducible germ of curve – on a normal surface singularity
X
X
. If
A
,
B
A,B
are two other branches, define
u
L
(
A
,
B
)
:=
(
L
⋅
A
)
(
L
⋅
B
)
A
⋅
B
u_{L}(A,B) := \dfrac {(L \cdot A) \> (L \cdot B)}{A \cdot B}
, where
A
⋅
B
A \cdot B
denotes the intersection number of
A
A
and
B
B
. Call
X
X
arborescent if all the dual graphs of its good resolutions are trees. In a previous paper, the first three authors extended a 1985 theorem of Płoski by proving that whenever
X
X
is arborescent, the function
u
L
u_{L}
is an ultrametric on the set of branches on
X
X
different from
L
L
. In the present paper we prove that, conversely, if
u
L
u_{L}
is an ultrametric, then
X
X
is arborescent. We also show that for any normal surface singularity, one may find arbitrarily large sets of branches on
X
X
, characterized uniquely in terms of the topology of the resolutions of their sum, in restriction to which
u
L
u_L
is still an ultrametric. Moreover, we describe the associated tree in terms of the dual graphs of such resolutions. Then we extend our setting by allowing
L
L
to be an arbitrary semivaluation on
X
X
and by defining
u
L
u_{L}
on a suitable space of semivaluations. We prove that any such function is again an ultrametric if and only if
X
X
is arborescent, and without any restriction on
X
X
we exhibit special subspaces of the space of semivaluations in restriction to which
u
L
u_{L}
is still an ultrametric.