  In this paper, we study the root distribution of some univariate polynomials
$W_n(z)$ satisfying a recurrence of order two with linear polynomial
coefficients over positive numbers. We discover a sufficient and necessary
condition for the overall real-rootedness of all the polynomials, in terms of
the polynomial coefficients of the recurrence. Moreover, in the real-rooted
case, we find the set of limits of zeros, which turns out to be the union of a
closed interval and one or two isolated points; when non-real-rooted polynomial
exists, we present a sufficient condition under which every polynomial with $n$
large has a real zero.
