I agree about the tree at Butler National ruining the hole. I played there a few times in college and thought that the trees really screwed that hole up. Trees on the inside of the hole and the grove at the end of the fairway mean that on a very long hole (450+ back in the late 80's) you had to hit a driver far, but not too far as you be stymied. The way the trees played it wasn't just one but several that could block you.
I've played a lot at Winged Foot but I fail to see how the tree on 10 East "made" the hole a good one. No doubt it was a beautiful tree but on a short hole where you would hit 3w and wedge a tree hanging over/onto the green seems to be very gimmicky to me. Were you supposed to lay up to have a longer iron in that you could keep under the limbs? The green was not receptive to bouncing a ball on so what were you supposed to do? Yes, it was a great tree--just in the wrong place.
Baltusrol (lower) #5 had a nice tree that would be worth saving. Also on the upper course I think on hole #1 there were some trees (or a tree) that forced a drive to be in the correct position.
At my home course there is a tree on the right of the fairway on #12 at The Honors Course that really narrows up the drive zone. Losing it would take this short hole (350 yards) and make it defenseless.