Let F be an algebraically closed field of characteristic 0 and f(x) a polynomial of degree strictly greater than one in F[x]. We show that the number of degree k polynomials with coefficients in F that commute with f (under composition) is either zero or equal to the number of degree one polynomials with coefficients in F that commute with f. As a corollary, we obtain a theorem of E. A. Bertram characterizing those polynomials commuting with a Chebyshev polynomial.