Dealing with Bradford Pear Tree Suckers
The Bradford pear tree has roots that tend to run along the surface of the ground. There is not a lot you can do about this, but you can try to keep the exposed roots covered. Most sucker growth starts where the roots are exposed to the air.
If the roots are exposed you can cover them with about 3 inches of soil or lawn sand. Grass will quickly grow through the layer or you can include some grass seeds in the cover when you apply it. Even though this will not stop suckers from growing, it will slow them down a bit. You will find that the roots will continually rise through the cover. This is a recurring problem and you will need to replace the cover each time it happens.
Dealing With Suckers
The only way to deal with the suckers is to cut them out as they appear. Do not cut out the roots that have the most suckers. You may think this is a simple solution, but cutting out these roots would do serious damage to the tree. In the past, some people have experimented with chemicals designed to inhibit the growth of suckers, but with no success.
The Bradford pear is a good shade tree, but understand that the suckers are a quirk you will encounter on a regular basis.