Tree Trimming Question?

[[JiF]KellysHero]

We have a tree that branches off into a Y, with the one part of the Y hanging over the house. Diameter of the both Y's is about 10 inches. This past winter, we had a major ice storm and the Y part was close to, if not touching the house. Tree trimmers are here, and curious if it would be best to cut it off at the base of the Y (just the part leaning over the house), or say halfway up the Y part that is leaning over the house. What is best for the life of the tree? Here are a couple of pictures of the tree.

[[JiF]Uncle Stinky]

OK, I must admit that my first instinct was to aim my SMAW rocket about halfway up the trunk on the right, and then blast it off. That way, I can spot the snipers on your roof more easily.

[[JiF]KellysHero]

Had two maple trees that were dying off from a drought 2 years ago, and/or some sort of disease. No way I'd be climbing up a tree like that!

[[JiF]Lt Gav]

I know that when the tree chaps do ours around the work site they would cut the branch of the trunk part way back but not right down.
The trees then sprout new growth from that side.
Your only problem then would be new growth + leaves much closer to the house so giving light issues.