  We define a notion called leftmost separator of size at most $k$. A leftmost
separator of size $k$ is a minimal separator $S$ that separates two given sets
of vertices $X$ and $Y$ such that we "cannot move $S$ more towards $X$" such
that $|S|$ remains smaller than the threshold. One of the incentives is that by
using leftmost separators we can improve the time complexity of treewidth
approximation. Treewidth approximation is a problem which is known to have a
linear time FPT algorithm in terms of input size, and only single exponential
in terms of the parameter, treewidth. It is not known whether this result can
be improved theoretically. However, the coefficient of the parameter $k$ (the
treewidth) in the exponent is large. Hence, our goal is to decrease the
coefficient of $k$ in the exponent, in order to achieve a more practical
algorithm. Hereby, we trade a linear-time algorithm for an $\mathcal{O}(n \log
n)$-time algorithm. The previous known $\mathcal{O}(f(k) n \log n)$-time
algorithms have dependences of $2^{24k}k!$, $2^{8.766k}k^2$ (a better analysis
shows that it is $2^{7.671k}k^2$), and higher. In this paper, we present an
algorithm for treewidth approximation which runs in time
$\mathcal{O}(2^{6.755k}\ n \log n)$,
  Furthermore, we count the number of leftmost separators and give a tight
upper bound for them. We show that the number of leftmost separators of size
$\leq k$ is at most $C_{k-1}$ (Catalan number). Then, we present an algorithm
which outputs all leftmost separators in time
$\mathcal{O}(\frac{4^k}{\sqrt{k}}n)$.
