  A graph that contains a spanning tree of diameter at most $t$ clearly admits
a tree $t$-spanner, since a tree $t$-spanner of a graph $G$ is a sub tree of
$G$ such that the distance between pairs of vertices in the tree is at most $t$
times their distance in $G$. In this paper, graphs that admit a tree
$t$-spanner of diameter at most $t+1$ are studied. For $t$ equal to 1 or 2 the
problem has been solved. For $t=3$ we present an algorithm that determines if a
graph admits a tree 3-spanner of diameter at most 4. For $t\geq4$ it is proved
that it is an NP-complete problem to decide whether a graph admits a tree
$t$-spanner of diameter at most $t+1$.
