  Let $G$ be a simple connected graph. For any two vertices $u$ and $v$, let
$d(u,v)$ denote the distance between $u$ and $v$ in $G$, and let $diam(G)$
denote the diameter of $G$. A radio-labeling of $G$ is a function $f$ which
assigns to each vertex a non-negative integer (label) such that for every
distinct vertices $u$ and $v$ in $G$, it holds that $|f(u)-f(v)| \geq diam(G) -
d(u,v) +1$. The span of $f$ is the difference between the largest and smallest
labels of $f(V)$. The radio number of $G$, denoted by $rn(G)$, is the smallest
span of a radio labeling admitted by $G$. In this paper, we give a lower bound
for the radio number of the Cartesian product of two trees. Moreover, we
present three necessary and sufficient conditions, and three sufficient
conditions for the product of two trees to achieve this bound. Applying these
results, we determine the radio number of the Cartesian product of two stars as
well as a path and a star.
