Ask Mr. Smarty Plants

We don't want to be flip, but have you considered a large umbrella?

Roots of trees generally extend at least as far out as the shadow of the canopy of the tree. This area is referred to as the Protected Root Zone (PRZ). While a tree and its rootball would probably initially fit into your 6' x 6' planter box, the roots will quickly grow beyond it.  Additionally, the roots of the tree have the potential to damage/crack any concrete they encounter as they grow trying to reach water and oxygen.

We are frequently asked for "taproot trees," on the assumption that a root that grows straight down will not interfere with concrete walks, driveways, sidewalks and foundations. Alas, some trees do begin with a taproot but there are few trees with a true taproot; as time goes by roots will spread from that center root, both in search of water and nutrients and also as a base to stabilize the tree in the ground. A tree tall enough, and with comparable width to provide shade for your patio is going to have roots extending far beyond the initial planter, if either the tree or the pool survive the planting there. A tree big enough to cast shade, say, 20 ft. tall, will usually have about the same spread-20' wide. The roots beneath that tree will normally spread out from two to three times farther than the crown.

There are shrubs that might fit into that 6x6 space and not interfere with your pool, but they will not get big enough to cast appreciable shade.