The City does not normally repair root damage on private property. However, if the City has erroneously planted a tree right on top of a sewer line, and the roots are interfering with sewage drainage, they will replace the tree. The City will repair the part of the sewer pipe that's in the public right of way, which is usually streetward of the cleanout. From the cleanout to the house is the owner's responsibility. Back to roots in lawns and gardens.
Big trees have big root systems, usually extending at least as far from the trunk as the canopy.
If you have a 10x20 rectangular lawn next to a tree, the tree will grow a 10x20 rectangular root system under the lawn because that's where the water is. Roots prefer loose soil to packed. Root patterns also vary by species. Magnolias and liquidambars are adapted to wetland conditions where there's little oxygen in the soil. Because roots need oxygen, there's no place for them to grow except along the surface.