Better sized pic here:

http://i.imgur.com/9yKxd.jpg

Pokeweed (Phytolacca) is a creeping perennial that's going to be tough to battle tangled up with tree roots, probably making digging it up impossible. If you keep cutting it down faster than it can grow, you will win eventually. You may want to try smothering the area with cardboard, which will allow water to get to the tree but block the pokeweed from getting the light it needs to survive. It looks like a mulched area anyway, so just cover the cardboard with mulch so you don't have to look at it, and make it heavy enough to suppress the pokeweed.

Pull any new sprouts you see right away, while they're easy. As you've seen, once huge roots develop, you've got a battle when it's time for it to go. It will just keep increasing its' spread every year if not killed. A cool-looking plant but not at all well-behaved enough for most people's yards. If it makes you feel any better, it IS a native.

Thanks so much, purpleinopp. At least now I have an ID and a plan.

I put a weekly alarm into my phone reminding me to cut the pokeweed down. ;)

We'll see how that goes for a while.

OP said "We'll see how that goes for a while."

Um, probably take several years. If you take a break, the plant will regain strength and you'll be back at square one.

Remove the green stuff every time you walk past it. Depending upon how rapidly it grows, once a week may be too seldom.

Your goal is to starve the fleshy root. You can do that only if you are persistent in removing all the green.