  Studying geodesics in Cayley graphs of groups has been a very active area of
research over the last decades. We introduce the notion of a uniquely labelled
geodesic, abbreviated with u.l.g. These will be studied first in finite Coxeter
groups of type $A_n$. Here we introduce a generating function, and hence are
able to precisely describe how many u.l.g.'s we have of a certain length and
with which label combination. These results generalize several results about
unique geodesics in Coxeter groups. In the second part of the paper, we expand
our investigation to infinite Coxeter groups described by simply laced trees.
We show that any u.l.g. of finite branching index has finite length. We use the
example of the group $\widetilde{D}_6$ to show the existence of infinite
u.l.g.'s in groups which do not have any infinite unique geodesics. We conclude
by exhibiting a detailed description of the geometry of such u.l.g.'s and their
relation to each other in the group $\widetilde{D}_6$.
