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.