It is difficult to say what to do about the tree without knowing for sure what type it is. Your description of its for them does leave room for it being a birch tree. I suggest you call your local County Extension Office to see about bringing in a photo or sample from the leaves so they can identify it for you. Their contact information is:
St. Johns County Extension, 3125 Agricultural Center Drive, St. Augustine, FL 32092-0572
Phone: (904) 209-0430
I hate to recommend removing a tree without knowing for sure what species it is. Birches suffer in hot summer weather especially and dry conditions.
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