Ask Mr. Smarty Plants

From a previous Mr. Smarty Plants answer:

"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). Covering this area with concrete is likely to cause the death of the tree by physically damaging the roots and by inhibiting oxygen and water from reaching the roots. Additionally, the roots of the tree have the potential to damage/crack the concrete 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 patio, if either the tree or the patio survive the planting there.