  We give a representation-theoretic bijection between rooted labeled forests
with $n$ vertices and complete exceptional sequences for the quiver of type
$A_n$ with straight orientation. The ascending and descending vertices in the
forest correspond to relatively injective and relatively projective objects in
the exceptional sequence. We conclude that every object in an exceptional
sequence for linearly oriented $A_n$ is either relatively projective or
relatively injective or both. We construct a natural action of the extended
braid group on rooted labeled forests and show that it agrees with the known
action of the braid group on complete exceptional sequences. We also describe
the action of $\Delta$, the Garside element of the braid group, on rooted
labeled forests using representation theory and show how this relates to
cluster theory.

Representation Theory;Combinatorics