  A tree $t$-spanner $T$ of a graph $G$ is a spanning tree of $G$ such that the
distance in $T$ between every pair of verices is at most $t$ times the distance
in $G$ between them. There are efficient algorithms that find a tree $t\cdot
O(\log n)$-spanner of a graph $G$, when $G$ admits a tree $t$-spanner. In this
paper, the search space is narrowed to $v$-concentrated spanning trees, a
simple family that includes all the breadth first search trees starting from
vertex $v$. In this case, it is not easy to find approximate tree spanners
within factor almost $o(\log n)$. Specifically, let $m$ and $t$ be integers,
such that $m>0$ and $t\geq 7$. If there is an efficient algorithm that receives
as input a graph $G$ and a vertex $v$ and returns a $v$-concentrated tree
$t\cdot o((\log n)^{m/(m+1)})$-spanner of $G$, when $G$ admits a
$v$-concentrated tree $t$-spanner, then there is an algorithm that decides
3-SAT in quasi-polynomial time.
