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Let $\mathbb{K}$ be a field and $S=\mathbb{K}[x_1,\dots,x_n]$ be the
polynomial ring in $n$ variables over the field $\mathbb{K}$. Let $G$ be a
forest with $p$ connected components $G_1,\ldots,G_p$ and let $I=I(G)$ be its
edge ideal in $S$. Suppose that $d_i$ is the diameter of $G_i$, $1\leq
i\leq p$, and consider $d =\max\hspace{0.04cm}\{d_i\mid 1\leq i\leq p\}$.
Morey has shown that for every $t\geq 1$, the quantity $\max\big\{\left
\lceil\frac{d-t+2}{3}\right\rceil+p-1,p\big\}$ is a lower bound for ${\rm
depth}(S/I^t)$. In this paper, we show that for every $t\geq 1$, the
mentioned quantity is also a lower bound for ${\rm sdepth}(S/I^t)$. By
combining this inequality with Burch's inequality, we show that any
sufficiently large powers of edge ideals of forests are Stanley. Finally,
we state and prove a generalization of our main theorem.
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