Quite opposite of the typical instinctive thought, putting a patio around a tree can be very beneficial to the tree. I will cover the fundamental benefit the tree receives from a patio and discuss the pros and cons of different construction materials / techniques.
The significant benefit that a tree gets from a patio is reduced soil compaction to its root zone. If there is no buffer between your feet and the soil, then the compaction that takes place over years of foot traffic becomes a very real stress factor to the tree. A patio can either reduce soil compaction by distributing the weight of your foot steps or eliminate compaction entirely by getting you completely off the ground.
Pavers are a popular choice for construction materials. They do a good job of letting soil and air penetrate the soil and do a fair job of distributing weight load to reduce soil compaction. The bigger each individual paver, the better job it will do distributing weight.
Concrete slabs do a much better job of distributing weight. Because the slab is one big piece, each foot step gets distributed over a larger area. Concrete becomes a problem in two scenarios. One, if you are covering more than half of the root system you might run into water and air exchange issues. And, if your slab requires you dig footing trenches around the edges the trenching might cut roots. However, if you slab is covering a small portion of the trees root system, then you can actually help water retention (remember that trees roots go out 4-5 times as far as the branches). Whenever it rains, the water moves laterally once it gets in the soil. So, water does get up under the concrete. However, as the sun and air dry up the water, the concrete slab will protect the soil underneath and act as an oasis of moisture after exposed soil has long since dried. I have heard and read about concrete buffer soil pH to deadly high ranges, but I have never experienced this on my projects.
The best patio construction material to use around trees is wood. Elevated wood decks get footsteps completely of the soil. And, if you provide small gaps between the boards, water and oxygen will easily access the soil and roots. In addition, the trees leaves will be able to slip through the cracks to the soil surface and retain a natural litter layer that recycles nutrients back into the soil. This is, hands down, the best way to go.
Whatever your approach, remember to leave enough room around the trunk for growth. You don’t want your patio girdling the tree trunk. And, if you have to dig piers or footers be careful. So, carry on, be careful and rest assured you are doing your tree a favor.