runner roots white poplar

I have a beautifull white poplar close to my kitchen garden.

The tree sends out a lot of runner roots in my kitchen garden. Shoots pop up everywhere (in the greenhouse, in my raised beds, from under the pavement. When I cut them off they grow back even harder. How can I get rid of them in an organic way whithout harming the tree. 

Your best bet is probably to tear the shoots off the root, not cut them. If you cut them, you leave a lot of nascent growth points behind and they will simply grow again.