  In this paper, we prove the real-rootedness of two classes of generalized
Narayana polynomials: one arising as the $h$-polynomials of the generalized
associahedron associated to the finite Weyl groups, the other arising in the
study of the infinite log-concavity of the Boros-Moll polynomials. For the
former, Br\"{a}nd\'{e}n has already proved that these $h$-polynomials have only
real zeros. We establish certain recurrence relations for the two classes of
Narayana polynomials, from which we derive the real-rootedness. To prove the
real-rootedness, we use a sufficient condition, due to Liu and Wang, to
determine whether two polynomials have interlaced zeros. The recurrence
relations are verified with the help of the Mathematica package
\textit{HolonomicFunctions}.
