  Tree spanners approximate distances within graphs; a subtree of a graph is a
tree $t$-spanner of the graph if and only if for every pair of vertices their
distance in the subtree is at most $t$ times their distance in the graph. When
a graph contains a subtree of diameter at most $t$, then trivially admits a
tree $t$-spanner. Now, determining whether a graph admits a tree $t$-spanner of
diameter at most $t+1$ is an NP complete problem, when $t\geq 4$, and it is
tractable, when $t\leq 3$. Although it is not known whether it is tractable to
decide graphs that admit a tree 3-spanner of any diameter, an efficient
algorithm to determine graphs that admit a tree 3-spanner of diameter at most 5
is presented. Moreover, it is proved that if a graph of diameter at most 3
admits a tee 3-spanner, then it admits a tree 3-spanner of diameter at most 5.
Hence, this algorithm decides tree 3-spanner admissibility of diameter at most
3 graphs.
