Let $K$ be a number field. We will show that any bicritical polynomial $f(z) in K[z]$ is conjugate to a polynomial of the form $amathcal{B}_{d,k}(z) +c in K[z]$ where $mathcal{B}_{d,k}(z)$ is a normalized single-cycle Belyi map with combinatorial type $(d; d-k, k+1, d)$. We use results of Ingram to determine height bounds on pairs $(a,c)$ such that $amathcal{B}_{d,k}(z) +c$ is post-critically finite. Using these height bounds, we completely describe the set of post-critically finite cubic polynomials over $Q$, up to conjugacy over $Q$. We give partial results for post-critically finite polynomials over $Q$ of arbitrary degree $d>3$.