In 2004, Herzog, Hibi, and Zheng proved that a quadratic monomial ideal has a linear resolution if and only if all its powers have a linear resolution. We study a generalization of this result for square-free monomial ideals arising from facet ideals of a simplicial tree. We give a complete characterization of simplicial trees for which all powers of their facet ideal have a linear resolution. We compute the regularity of t-path ideals of rooted trees. In addition, 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 $\Delta $ is a d-dimensional simplicial tree connected in codimension one, then reg$(I(\Delta )^s) \leq (d+1)(s-1)~+$ reg$(I(\Delta ))$ for all $s \geq 1$. We prove this conjecture for some special classes of simplicial trees.

We investigate whether the property of having linear quotients is inherited by ideals generated by multigraded shifts of a Borel ideal and a squarefree Borel ideal. We show that the ideal generated by the first multigraded shifts of a Borel ideal has linear quotients, as do the ideal generated by the $k$th multigraded shifts of a principal Borel ideal and an equigenerated squarefree Borel ideal for each $k$. Furthermore, we show that equigenerated squarefree Borel ideals share the property of being squarefree Borel with the ideals generated by multigraded shifts.