Depth of powers of edge ideals of Cohen-Macaulay trees

Let $I$ be the edge ideal of a Cohen-Macaulay tree of dimension $d$ over a polynomial ring $S=\mathrm{k}[x_1,\dots ,x_d,y_1,\dots ,y_d]$. We prove that for all $t\ge 1$, $$\mathrm{depth}(S\mathrm{/}{I}^t)=\max \{d-t+1,1\}\mathrm{.}$$