  In this paper, we study dynamics of maps on quasi-graphs characterizing their
invariant measures. In particular, we prove that every invariant measure of
quasi-graph map with zero topological entropy has discrete spectrum.
Additionally, we obtain an analog of Llibre-Misiurewicz's result relating
positive topological entropy with existence of topological horseshoes.
  We also study dynamics on dendrites and show that if a continuous map on a
dendrite, whose set of all endpoints is closed and has only finitely many
accumulation points, has zero topological entropy, then every invariant measure
supported on an orbit closure has discrete spectrum.
