  Given a quantum cluster algebra, we show that its triangular bases defined by
Berenstein and Zelevinsky and those defined by the author are the same for the
seeds associated with acyclic quivers. This result implies that the
Berenstein-Zelevinsky's basis contains all the quantum cluster monomials.
  We also give an easy proof that the two bases are the same for the seeds
associated with bipartite skew-symmetrizable matrices.
