Most of a tree's roots are in the top 3 feet of soil, and half of the root system of most trees is less than a foot below the surface of the ground. These roots need access to air and water, and covering the majority of the root system with concrete so that water and air can't get to it can suffocate the tree. To avoid this problem, don't cover more than half the tree's root system with concrete, keeping in mind that the roots may extend up to five times as far from the trunk as the branches do.
Tree roots don't like to be disturbed, either, so excavating around the tree in preparation for paving can be problematic if it moves or displaces roots. The damage can be even worse if the excavation severs a significant number of roots, as often occurs when builders dig trenches for pavement or foundation footers. Some trees are more sensitive to root disturbance than others; pines, for example, can tolerate having some of their roots cut, while oaks are particularly sensitive to root damage.
Damage to Pavement
Although injury to a tree caused by nearby pavement is a concern, the damage can be inflicted in the other direction, too. Tree roots can grow under concrete pavement that is too close to the tree, and when frost heave or cracks in the pavement allow the roots to work their way into the concrete, the expanding roots can cause the pavement to buckle and break. You can repair the damage, but it's likely to recur; the best solution is to move the walkway or pavement further away from the tree.
In general, it's best to keep concrete walkways, pavement and foundations as far from trees as possible, and walkways should never be closer than three feet from the tree's trunk. If the tree in question can handle some root disturbance, a vertical root barrier in a trench between the tree and the pavement can prevent the roots from spreading under the pavement and damaging the concrete. Using alternative materials, such as a raised wooden deck, for landscaping features near trees helps, too.