Answer: It is very difficult to establish either trees or shrubs under the dripline and within the root zone of a large established tree because the competition is very fierce for sunlight, water, and nutrients. Furthermore, if you dig out a substantial strip of roots you risk damaging the tree. If you are determined to use shrubs, you might consider the shallow rooted rhododendrons as a possibility, although it may be quite difficult to get them established.
Since it sounds like you would really like to screen this area well, it might be worth considering an attractive trellis or fence and then adding a nice tracery of vine to the fence to soften the look of it. In this case, any one of several vines might be attractive. The deciduous Virginia Creeper will grow in shade and will cling by little aerial roots; it turns bright red in the fall. English ivy, also a shade grower, is an evergreen clinging vine and it is available in variegated shades as well as the plain dark green form. Another possibility would be the fall blooming, sweet autumn clematis which is a very sturdy vine although it looks quite delicate; its white blooms are sweetly fragrant and would be a lovely addition to any garden. Finally, a small burbling fountain placed close to your seating area can work surprisingly well as a method of masking unpleasant neighboring noise.
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