I personally wouldn't put trees around a pool. The roots would grow into the decking / shell. If it's shade you need or privacy, put up some umbrellas. I'd plant shallow type plants that won't have a huge root system.
I like the idea of hollys -- They don't require a lot of care. There're different sizes types and those without thorns. They hold up well in the heat and full sun. You could go to youtube and look them up - There's quite a few horticultural sites and holly is very popular.
Texas Mountain Laurel. The purple flowers smell like grape bubble gum. They can be a hedge or a small tree. I overbought these at Lowe's last year. I think I have six right now. They are slow growing.
One I'm thrilled with is Carolina Sapphire Arizona Cypress. With Christmas approaching and box stores putting out small live 'Christmas' trees, you'll find various types like this. It's hardy, does well in all soil types, withstands the heat, and has this awesome blue color. There's another one called 'Blue Ice Cypress.' The main thing is it needs to drain well so elevate the root ball above the soil level. Make a mound!
We've had great luck with Texas Sage. It's that gray plant that blooms small purple flowers when it rains. They can grow into a hedge or cut back. The root system isn't extensive.
I would say a big no on all the palm trees. They don't scream tropical to me. On some the seeds are poisonous.
Hibiscus - Unless you have an extremely green thumb and know what you're doing, skip this one. They don't do well in a freeze.
Butterfly bushes are colorful, fun, and fast growing. But they attract bees and butterflies and shed. This is really easy to grow.
No on Oleander. They're extremely poisonous to dogs and kids. All parts of the plant are poisonous.
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I cut them down 3 weeks ago. While they survived the quick snow of 2017 (pictured below), the week-long freeze of 2021 was too much. They thinned-out and lost so much needles that my skimmer had to be emptied at least twice a day. A complete & total mess. I'll never plant a tree near a pool again. Debris and root systems are potential problems. Not always mentioned, but trees attract birds and that causes more of a mess at the pool. There are a variety of evergreen bushes and shrubs as noted above, so I would recommend those if needed, but that's about it.
