  A radio labelling of a graph $G$ is a mapping $f : V(G) \rightarrow \{0, 1,
2,\ldots\}$ such that $|f(u)-f(v)|\geq diam(G) + 1 - d(u,v)$ for every pair of
distinct vertices $u,v$ of $G$, where $diam(G)$ is the diameter of $G$ and
$d(u,v)$ is the distance between $u$ and $v$ in $G$. The radio number $rn(G)$
of $G$ is the smallest integer $k$ such that $G$ admits a radio labelling $f$
with $\max\{f(v):v \in V(G)\} = k$. In this paper, we give a lower bound for
the radio number of the Cartesian product of a tree and a complete graph and
give two necessary and sufficient conditions to achieve the lower bound. We
also give three sufficient conditions to achieve the lower bound. We determine
the radio number for the Cartesian product of a level-wise regular trees and a
complete graph which attains the lower bound. The radio number for the
Cartesian product of a path and a complete graph derived in [Radio number for
the product of a path and a complete graph, J. Comb. Optim., 30 (2015),
139-149] can be obtained using our results in a short way.
