Question
My husband and I have a tree problem...actually, our neighbor has a tree problem but doesn't see it as such!  Right next to our fence is the neighbor's tree.  This summer, we have now noticed we have several (read tens) of little tree sprouts in our grass that are growing rapidly.  Our grass this year is also very brown in one spot, regardless of where the sprinkler systems sprays. Where when my husband dug up one of the sprouts in this brown spot, a massive root system was discovered.  What can we do, short of cutting down the tree?

Answer
The sprouts are coming from the root system of the tree. You did not say what species of tree was involved so i am going to give some general statements.
Is the tree beneficial to you and your yard--shade. etc.?
The tree's roots are taking the moisture that the grass would have gotten. That makes the browning of the grass.
You can add soil to this area (about 4 inches maximum) and then grow the grass. Or grow a ground cover besides grass in this area.
You can just add the top soil and  just keep mowing down the sprout as they come up.   
You can wipe or spray the sprouts with a herbicide called Roundup. Spraying will kill the green plants in then area including the grass. Wiping the leaves of the sprouts with
the Roundup will kill the sprouts. This may also damage some of the foliage of the tree on that side since the roots will take the herbicide up to the leaves.
You can also trench along the fence and install a root barrier to keep the roots from coming on your side -this may also cause some damage to the foliage of the tree.

I think I would wipe the sprouts with Roundup and add top soil and replant the grass OR maybe put a different ground cover under the tree.