Answer: In my experience, this is a case of "If you can't beat 'em, join 'em!" Such a tree really presents a difficult situation. Unfortunately, as you have discovered, the tree is truly a water hog and easily outcompetes most other plants. While you may have some temporary successs by digging out individual planting pockets, in the long run the roots will regrow and cause trouble again -- sometimes within months. In this case I would suggest establishing a strenuous ground cover such as English ivy and then using some large containers in which to grow annuals for color. Since this is a large area you might consider using some of it for an informal path and clearing around a welcoming bench located in that cool shady spot, perhaps even adding a little fountain along with the planters.
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