Reclaiming a Tree From Ivy

Behind my new house (zone 6) there is a huge maple, probably 5-6 feet in diameter, which the previous owners allowed to be swamped by english ivy.
It had gotten probably 20 feet up the tree, and some of the vines were two inches in diameter.
I have cut the ivy down and pulled up as many roots as possible (realizing that new shoots may be coming back for some time). Should I do anything else to help the tree?
Some of the bark at the base of the tree appears to have been rotted where the thickest vines started climbing, although the rot doesn't appear to be deep or wide.
The roots were getting swamped by a dense mat of ivy roots, runners, and thatchy pulp about 6 inches deep. I know that in general trees don't like additional cover over their roots.
I'm guessing that from time to time someone went over the patch with a lawnmower but did nothing to actually uproot the ivy. I've pulled out the mat about a foot from the trunk but I'm wondering how much more I should yank it out -- it extends probably 4 feet from the trunk, so I'm guessing it covers 100+ square feet.
Would it make more sense to just cut and yank out the ivy as it reappears and let the roots starve to death, then let the mat slowly decompose, or is more active removal wise? I know that ivy roots are very resistant to rot, so I'm reluctant to let nature run its course if that means smothering the roots of this tree.
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The tree is 5-6 feet in diameter. The ivy is in a ring only 13-14 feet in diameter (4 feet outside the tree trunk). The drip line is much bigger than this. I doubt that the ivy is interfering with the roots much any more. Probably just mowing it would keep the ivy in check and keeping it away from the bark will keep the tree healthy.
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Thanks for the comment.
Any suggestions on how to turn this tangled mat of roots and leaves and stems and who knows what else into something plantable without pulling it up? Will mulching over it help speed the killing and decomposition process, or will it just give the ivy a better environment for spreading?
I'd like to put some kind of shade-tolerant non-invasive ground cover in place of the ivy, or even just mulch, something to cover up this ugly thicket. But I don't want to do anything until I know I have the ivy under control (I know it's not easy to kill for good).
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Not much grows under Norway Maples. If it is some other kind of Maple, you may have better luck.
I assume by ivy, you mean English Ivy, Hedera helix. It is considered "a noxious invasive plant." Washington State University [http://www.nwcb.wa.gov/weed_info/Ivy.htm ] recommends:
1) Burning plants and re-sprouts at regular intervals with a blow torch will eventually deplete the plant's energy.
2) Due to its waxy cuticle, ivy is not easily controlled with herbicides. However, Derr reported complete control of English ivy with two applications of 2,4-D at 1 lb/acre, applied as foliar sprays in June and August.
After doing either 1) or 2) above, I would recommend covering it with black plastic and covering that with a good mulch. Then after a couple years, it should be OK to plant. You might have to repeat 1) or 2) on the edges periodically.
Note: Norway Maple is also considered "a noxious invasive plant."
Also, check out: http://www.nps.gov/plants/alien/fact/hehe1.htm
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