I have spend quite a bit of money removing ivy which engulfed a eucalyptus tree (before I bought my house). And I still have lingering issues.

One minor issue is fighting the ivy coming from my invasive-plant loving neighbor <insert Round-up + clipping>. The major issue is that there are still ivy "stumps" at the base of the tree which are thick & intwined yet mostly underground.

All internet references to ivy relate to leaves, vines, etc. But what about the remaining thick root systems which are underground, and have no leaves? I am finding white/purple like shoots underground.