Answer: Morus (mulberry) has a reputation for developing above-surface roots. They are wonderful shade trees, though, and provide a dense umbrella of welcome shade during hot summer months.
Just a word of caution: Any tree planted in a lawn that's regularly watered and fertilized will tend to produce a lot of surface roots. As trees mature these roots can rise several inches above ground level, tripping people and causing hazards for lawnmowers.
A few of the more well-behaved deciduous trees that you might consider include Acer griseum (paperbark maple), Cercis (redbud), Chilopsis linearis (desert willow) and Stewartia.
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