The first step to repairing the damage is to remove the sections of pavement that are being raised or cracked. Removing sections can be difficult, but an asphalt or concrete saw will work. These tools will allow you to remove the broken pieces of pavement and get down to the problem.
Don’t repave the area yet, because the problem will return. Instead, you want to remove any small roots that you find in the area. You can dig down under the asphalt and cut the roots. If the roots are large, don’t cut them, as this could lead to the tree dying, which would cause even more trouble.
What you want to do in this case is to install a root shield, which will prevent the small roots from regrowing. You may also be able to dig down underneath the roots, which will encourage them to grow deeper and not damage your pavement.
Once you’re done with that, the area can be repoured or patched. You can do this by applying fresh asphalt or concrete to the area. The only downside to patching asphalt or concrete is that not everyone has the tools, time, or knowledge to do the job right.