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A tree $t$-spanner $T$ of a graph $G$ is a spanning tree of $G$ such that thedistance in $T$ between every pair of verices is at most $t$ times the distancein $G$ between them. There are efficient algorithms that find a tree $t\cdotO(\log n)$-spanner of a graph $G$, when $G$ admits a tree $t$-spanner. In thispaper, the search space is narrowed to $v$-concentrated spanning trees, asimple family that includes all the breadth first search trees starting fromvertex $v$. In this case, it is not easy to find approximate tree spannerswithin 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 receivesas 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 decides3-SAT in quasi-polynomial time.

On approximating tree spanners that are breadth first search trees