Can he sever the roots along the property line?
What if the trunk is on the property line?
In Illinois, if the tree is growing on your property, it belongs to you. You own the trunk, roots and branches. If you want to remove it, that is your right. If you want to keep it, that is your right too.
Branches: If your neighbor doesn't like the branches that are hanging over his yard, he needs to have your permission before he removes them or prune them back in a way that could harm the tree's health or change its shape (unless the tree or branches are considered hazardous and dangerous - check with your community's code enforcement staff to find out how they define "hazardous and dangerous" trees).
Further, you are not required to pay for trimming or pruning a tree that hangs over the neighbor's property, unless it is considered hazardous and dangerous.
Roots: A neighbor needs your permission if he is going to trench or cut through the roots of your tree, even if he is trenching in his yard. An exception would be if he can show that the roots are damaging his property (maybe harming the foundation of his home or clogging his sewer line).
Property line: If the tree is on the property line, you own it equally, which means that one neighbor cannot unilaterally take down the tree without permission of the other. nor can one neighbor legally sever or damage the root system on his property without your permission.
Recently, someone told me about a situation where there was a pine tree on the property line. One neighbor hated the tree, so he cut off all the branches on his side of the tree and then severed the roots on his side of the lot line. Shortly afterward, there was a storm and the weakened tree fell over - onto the house of the guy who damaged it!
Here's a nice article from the Village of Northfield that might be of interest.
If dead branches from the tree fall on the neighbor's property, you are not obligated to remove and dispose of them (but it might be the neighborly thing to do), just as you are not under any obligation to rake the leaves from your tree that fall on another's property! Frankly, both situations are part of living in a community.
We all have a responsibility to respect the rights of others and their property.