  Exceptional sequences are certain ordered sequences of quiver
representations. We use noncrossing edge-labeled trees in a disk with boundary
vertices (expanding on T. Araya's work) to classify exceptional sequences of
representations of Q, the linearly-ordered quiver with n vertices. We also show
how to use variations of this model to classify c-matrices of Q, to interpret
exceptional sequences as linear extensions, and to give a simple bijection
between exceptional sequences and certain chains in the lattice of noncrossing
partitions. In the case of c-matrices, we also give an interpretation of
c-matrix mutation in terms of our noncrossing trees with directed edges.
