It sounds like the tree could be sufferring from 2 different problems.

An Iron deficiency can cause yellowing of the leaves,but don't know if it would cause the type of dieback that you are experiencing.

An infestation of the Bronze Birch Borer does cause dieback. If it is caught and treated in time you shouldn't lose the tree.

Here's some info regarding prevention and treatment for the Borers.....
Bronze Birch Borer/US extension office

Hope this helps and Welcome to the Forum

G-Mom

Birches are also host to leaf miners look at the leaves and see if they appear to be hollowed out.
A lot of the time you will see what appears to be dirt on the inside of the leaf, this is actually the fras or "poop" of the little critters. This remains after the leaf miners morph from their larva stage and leave the leaf. If it is a leaf miner pretty much the same treatment for the birch borer will work. (a ground drench of Merit) You will need to treat the tree systemically as the critters are inside the leaf. It is best to do the application in the late fall as the trees are soaked with sap in the early spring.

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Mike Maier
aka
The Plant Doc