You can cut away a certain amount of roots and the tree will just grow more. Just how much is hard to say. I am not sure I would take the chance since the sidewalk must be close to the base of the tree and it would mean a major cutting job. You might look at options such as replacing the concrete with asphalt or brick than can give with the growth of the tree. Could you put in a short raised boardwalk? Perhaps even reroute the side walk. I have seen them arcing around the tree. Unfortunately, you may need to work with short sighted city officials on a front sidewalk. They are too quick to take the easy way out and cut the tree down. The federal government seems to have an urban clear cut policy. Stand up for your tree.
The tree has an economic value. I am sure the price you paid for the house reflected it, and when time comes to sell, you can sell for more if you have accommodated the tree. If it shades the house in the summer, it can reduce air conditioning bills or even avoid having air conditioning. Better to put the money into replacing the sidewalk a few times than air conditioning. Removing the tree won't be cheap either.
Nothing adds to the character of an older house like mature trees. They just don't look right in the sun with a little sapling out front.
If you want, you can email the pictures to
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.