  Let u be a cyclic word in a free group F_n of finite rank n that has the
minimum length over all cyclic words in its automorphic orbit, and let N(u) be
the cardinality of the set {v: |v|=|u| and v=\phi(u) for some \phi \in AutF_n}.
In this paper, we prove that N(u) is bounded by a polynomial function of degree
2n-3 with respect to |u| under the hypothesis that if two letters x, y occur in
u, then the total number of x and x^{-1} occurring in u is not equal to the
total number of y and y^{-1} occurring in u. We also prove that 2n-3 is the
sharp bound on the degree of polynomials bounding N(u). As a special case, we
deal with N(u) in F_2 under the same hypothesis.
