[header] How to kill aspen shoots [title] Treat the shoots in late summer or early fall, if you can. [step] Do this before the leaves change color. Herbicides are the best way to eliminate the shoots, though they will also kill the original tree. 
It takes several months for the herbicide to work. The best time to use it is right before the weather starts to cool down, since the tree will be in a dormant period anyway.