I have a poplar tree right behind my garden. Its invasive root system makes nothing grow in my garden except for some weeds. I recently dug out one of its woody anchor roots plus some of the smaller transport and feeder roots in my garden, about 1/8 of its root area. I intended to keep the tree alive. My question is two folds. Will its root system invade my garden again? Do I need to dig out the dirt between the poplar and my garden every year to check and cut any new roots?
Secondly, for those smaller roots remaining in my garden that originally connected to the cut anchor root, will they eventually die? Should I apply herbicide (such as Brush-B-Gon) over them? I worry that they may continue to live and hurt my future plants.
Thank you for any kind help and suggestions.