Aspen trees are relatively short lived which is why they routinely send up suckers, both to colonize an area but also to serve as replacements for the original tree. You can cut or pull the suckers up but there is no way to stop them from growing without compromising the health of the tree. If you were to apply Brush B Gon to the stubs after cutting back the suckers, the chemical could be carried to the roots of the tree and might kill it. The problem with that is that the tree would probably send up hundreds of new suckers while it was dying. A better approach would be to have the tree cut down and the stump and roots ground or dug out of the soil. Simply cutting the tree down will still leave roots in the landscape and those roots will probably send up suckers. So the bottom line is that there is no easy solution to your suckering aspen tree.
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