In the fall of 1968, Donald J. Trump received a timely diagnosis of bone spurs in his heels that led to his medical exemption from the military during Vietnam.

For 50 years, the details of how the exemption came about, and who made the diagnosis, have remained a mystery, with Mr. Trump himself saying during the presidential campaign that he could not recall who had signed off on the medical documentation.

Now a possible explanation has emerged about the documentation. It involves a foot doctor in Queens who rented his office from Mr. Trump’s father, Fred C. Trump, and a suggestion that the diagnosis was granted as a courtesy to the elder Mr. Trump.

The podiatrist, Dr. Larry Braunstein, died in 2007. But his daughters say their father often told the story of coming to the aid of a young Mr. Trump during the Vietnam War as a favor to his father.