  The Hitchin component is a connected component of the character variety of
reductive group homomorphisms from the fundamental group of a closed surface S
of genus greater than 1 to the Lie group PSL_m(R). The Teichmuller space of S
naturally embeds into the Hitchin component. The limit points in the Thurston
compactification of the Teichmuller space are well-understood. Our main goal is
to provide non-trivial sufficient conditions on a sequence of Hitchin
representations so that the limit of this sequence in the Parreau boundary can
be described as an action on a tree. These non-trivial conditions are given in
terms of Fock-Goncharov coordinates on moduli spaces of generic tuples of
flags.
