Powers of facet ideals of simplicial trees

In this article, we study the linearity of the minimal free resolution of powers of facets ideals of simplicial trees. We give a complete characterization of simplicial trees for which (some) power of its facet ideal has a linear resolution. We calculate the regularity of the $t$-path ideal of a perfect rooted tree. We also obtain an upper bound for the regularity of the $t$-path ideal of a rooted tree. We give a procedure to calculate the regularity of powers of facet ideals of simplicial trees. As a consequence of this result, we study the regularity of powers of $t$-path ideals of rooted trees. We pose a regularity upper bound conjecture for facet ideals of simplicial trees, which is as follows: if $\mathrm{\Delta }$ is a $d$-dimensional simplicial tree, then $\text{reg}(I(\mathrm{\Delta }{)}^s)\le (d+1)(s-1)+\text{reg}(I(\mathrm{\Delta }))$ for all $s\ge 1$. We prove this conjecture for some special classes of simplicial trees.