A non-toxic way to kill weeds/unwanted plants is boiling water. I have found it more effective than some of the vingegar & soap homemade weedkiller recipes out there, if you want to avoid Round-Up. Hopefully the tree roots would survive if the lilies-of-the-valley don't totally surround the tree...
I agree, it looks like Lily of the Valley. Does it have a nice smell? If it does I would be pretty sure that's what it is. It looks naturalized, so may be hard to fully remove, but it's poisonous enough I would fence off whatever you can't remove. All parts are poisonous, leaves, flowers and...