Removing trees often causes more damage than leaving them in situ. At present, as long as there is no obvious damage to walls etc, then the tree is removing a lot of water from the soil. When removed that water has to go somewhere. Also the roots of the removed tree would shrink and caused sinkage. If there is no problem then leave it alone.

OB is taking about 'heave' which as he says can be a problem when removing established trees. As I see it, you still have two alternatives. You can dig down about a few feet from your house and cut off any roots or gradually reduce the height and width of the tree by pruning over a few years.

Thanks for the advice, very gratefully received. It's nice to know I'm doing the right thing by leaving the tree there. I do have it pruned about every other year, by a tree surgeon, so that I can see the garden through that window!