  We investigate the syntactic complexity of certain types of finitely
generated submonoids of a free monoid. In fact, we consider those submonoids
which are accepted by circular semi-flower automata (CSFA). Here, we show that
the syntactic complexity of CSFA with at most one `branch point going in' (bpi)
is linear. Further, we prove that the syntactic complexity of $n$-state CSFA
with two bpis over a binary alphabet is $2n(n+1)$.
