Ask Mr. Smarty Plants

You're kidding, right? You don't really want something that's going to drop loads of leaves in your pool? There are two native mulberries, Morus microphylla (Texas mulberry) and Morus rubra (red mulberry). Regardless of whether it was native or not, you did the right thing to remove the tree. Whether it should be replaced by a new tree or not is another question. Roots of any tree and in-ground swimming pools are natural enemies, both for reasons you have already stated-invasive roots and messy leaf and fruit drop.  You didn't say how much distance between house and pool is involved. but consider this: If you move the tree away from the pool to protect the pool filters and appearance, the roots could very well interfere with the house foundation as well as walks, patios, etc. If you move it closer to the pool, you're right back where you started. Although a newly-planted tree would be small and not intrusive, it's going to grow. It won't be any good as a shade tree until it gets quite a bit bigger, an amount of time that depends on the speed with which the tree grows. And whether a tree is evergreen or deciduous, there is going to be constant shedding of dead leaves, little twigs, over-ripe fruit and your odd bird's nest. Although planting a tree might seem to be an inexpensive method to acquire shade, it will be a few years, at least, before you get the shade.

Although we will suggest some trees that are native to Nevada, we would urge you to consider alternatives, like a pavilion or patio cover for shade, using the the smaller trees for a softening effect. We have selected two smaller trees, more ornamental than shade trees, and two larger fast-growing trees. The majority of the larger trees suitable for shade are also known for surface roots which can buckle paving and interfere with other plantings. 

TREES

Cercis orbiculata (California redbud) - deciduous ornamental, blooms white, red, pink, purple March to May

Chilopsis linearis (desert willow) - deciduous, not actually related to true willows, white, pink, purple blooms April to September.

Fraxinus velutina (velvet ash) - deciduous shade tree, to 40 ft., fast-growing.

Pinus contorta (lodgepole pine) - evergreen, fast-growing, must be transplanted when small because of danger of damaging taproot.

Cercis orbiculata

Cercis orbiculata

Chilopsis linearis

Chilopsis linearis

Fraxinus velutina

Pinus contorta

Pinus contorta