If it was apple scab you would see mis-shapen fruit with big black blotches all over them, same with the leaves. So that isn't your problem. Is it just the one branch or all over the tree? If just the one branch, check the bark on the branch going to the stem. Sometimes a bad pruning cut can get infected with bacterial canker or other diseases and the bark around the branch will die, killing the branch.

I just checked the tree and it's not just the same branch. Although it is mostly located on the one branch, it appears it is also happening to a lesser degree in two or three other areas. The premature apple drop appears to be happening to the entire tree. We did have someone prune it last year, but I can't see any issues at any of the cuts. I'm glad it doesn't appear to be apple scab. Maybe it's a weather thing?

Is this possibly a rather old tree, maybe it's just approaching the end of its life span.

I'm not sure how old the tree is, but it's definitely mature. We moved in a couple years back and the tree was already here. I hope it's not nearing the end of it's life span, but I guess that may be a possibility. It does get a lot of suckers at the base that I have to trim down each year, so maybe that does signify poorer health?

It could have some disease or winter damage,..I would cut the branch back.

Thanks for the suggestions! We are going to cut that branch back this weekend. I noticed tonight that there is another branch coming off the same larger branch that doesn't have ANY leaves (I'm not sure if it ever did, or if they all yellowed as well and fell off). So we'll cut that whole piece out back to the main branch, and hopefully that will take care of most of it.

At this point, I wouldn't fret. Sometimes older branches just die as part of the tree's development, and newer growth in the crown or on other branches replaces the old growth. Prune it out and see how things go.

If you take a close look at any unpruned tree you will see some dead branches but mostly live branches. That's just how trees grow naturally. We like to prune out the dead branches to make the tree look its best, but a few dead branches does not by itself mean anything is wrong with the tree. If it did, I'd have to cut down that big oak in my backyard, which is showing some dead branches from last winter that I've yet to deal with (and may be too high up for me to ever deal with)....