  We present a new characterization of $k$-trees based on their reduced clique
graphs and $(k+1)$-line graphs, which are block graphs. We explore structural
properties of these two classes, showing that the number of clique-trees of a
$k$-tree $G$ equals the number of spanning trees of the $(k+1)$-line graph of
$G$. This relationship allows to present a new approach for determining the
number of spanning trees of any connected block graph. We show that these
results can be accomplished in linear time complexity.
