In the fall of 1968, Donald 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 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, New York, who rented his office from Trump’s father, Fred Trump, and a suggestion that the diagnosis was granted as a courtesy to the elder 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 Trump during the Vietnam War as a favor to his father.

“I know it was a favor,” said one daughter, Dr. Elysa Braunstein, 56, who along with her sister, Sharon Kessel, 53, shared the family’s account for the first time publicly when contacted by The New York Times.

Elysa Braunstein said the implication from her father was that Donald Trump did not have a disqualifying foot ailment. “But did he examine him? I don’t know,” she said.

For decades, Larry Braunstein saw patients in a congested ground-floor office below Edgerton Apartments in Jamaica, Queens, one of dozens of buildings owned by the Trumps in the 1960s. The family sold the building in 2004, records show.

“What he got was access to Fred Trump,” Elysa Braunstein said. “If there was anything wrong in the building, my dad would call and Trump would take care of it immediately. That was the small favor that he got.”

No paper evidence has been found to help corroborate the version of events described by the Braunstein family, who also suggested there was some involvement by a second podiatrist, Dr. Manny Weinstein. Weinstein, who died in 1995, lived in two apartments in Brooklyn owned by Fred Trump; city directories show he moved into the first during the year Donald Trump received his exemption.

Braunstein’s daughters said their father left no medical records with the family, and a doctor who purchased his practice said he was unaware of any documents related to Trump. Most detailed government medical records related to the draft no longer exist, according to the National Archives.

In an interview with The Times in 2016, Trump said that a doctor provided “a very strong letter” about the bone spurs in his heels, which he then presented to draft officials. He said he could not remember the doctor’s name. “You are talking a lot of years,” Trump said.

But he suggested he still had some paperwork related to the exemption, which he did not provide.

Trump did not mention in that interview any connection between his father and the doctor. The White House did not make Trump available for a follow-up interview and did not respond to written questions about his service record.

An investigation by The Times in October showed the extent to which Fred Trump had assisted his son over the years, despite Donald Trump’s insistence to the contrary. The investigation revealed that Donald Trump received the equivalent today of at least $413 million from his father’s real estate empire, including the equivalent of $200,000 a year by age 3.

In the 1960s, there were numerous ways to avoid military service, especially for the sons of wealthy and connected families, but Trump has said that no one pulled strings for him.

“I didn’t have power in those days,” Trump told biographer Michael D’Antonio in a 2014 interview, according to transcripts shared with The Times. “I had no power. My father was a Brooklyn developer, so it wasn’t like today.”

Dr. Alec Hochstein, who worked with Larry Braunstein in the late 1990s, said the podiatrist had recalled over dinner with their wives how the Trumps had treated him well, including backing off from rent increases. Hochstein did not remember any discussions related to Donald Trump’s medical exemption.

“He spoke very highly of the Trumps, and they were very open to negotiating with him and letting him stay in the space at a rent he was comfortable with,” Hochstein said.

Dr. Nicholas Campion, who bought Larry Braunstein’s practice around the time that the Trumps sold the building, which was less than a mile from the Trump family home in Jamaica Estates, said Donald Trump had had a large presence in the community.

“Everybody recalls the Trump family around Jamaica Estates,” Campion said.

In recent years, the diagnosis of bone spurs has subjected Trump to ridicule from critics, who have found it implausible that a healthy and athletic 22-year-old, on the cusp of being declared fit for service, could suddenly be felled by growths in his heels. Trump’s own shifting narrative over the years about his Vietnam-era experience has added to the suspicions.

At the time of the diagnosis, Trump was navigating a tumultuous period for the country after he graduated from the University of Pennsylvania. The United States inducted about 300,000 men into the military in 1968. At that time, a year before the draft lottery was instituted, local boards had to meet quotas and called men for service, leaving those without deferments or exemptions vulnerable.

Trump had been declared available for service two years earlier and had undergone a physical exam, Selective Service records show. That exam did not result in a medical exemption, but he did receive an education deferment. When officials again declared him available for service in July 1968, he had exhausted four education deferments and finished school, so it was the medical exemption that kept him from being eligible.

He has often said it was “ultimately” a high draft lottery number that spared him, but Trump had been medically exempted for more than a year before the lottery began in December 1969.

Beginning in October 1968, records show, Trump had a 1-Y classification, a temporary medical exemption, meaning that he could be considered for service only in the event of a national emergency or an official declaration of war, neither of which occurred during the conflict in Vietnam. In 1972, after the 1-Y classification was abolished, his status changed to 4-F, a permanent disqualification.

The Times began looking into Trump’s draft record anew when an anonymous tipster suggested that a podiatrist who was a commercial tenant of Fred Trump’s had provided the medical documentation.

The tipster offered no names, but The Times used old city directories, held by the New York Public Library, and interviews with Queens podiatrists to identify Larry Braunstein.

The doctor’s daughters said his role in Donald Trump’s military exemption had long been the subject of discussions among relatives and friends.

“It was family lore,” said Elysa Braunstein. “It was something we would always discuss.”

She said her father was initially proud that he had helped a “famous guy” in New York real estate. But later, her father, a lifelong Democrat who had served in the Navy during World War II, grew tired of Donald Trump as he became a fixture in the tabloid gossip pages and a reality television star, she said. The daughters, both Democrats, say they are not fans of Trump.

Trump has had a complicated relationship with the military, having quarreled with the likes of Sen. John McCain, a prisoner of war during Vietnam; the parents of a slain soldier; and the architect of the Osama bin Laden raid, even while speaking during campaign rallies about his enthusiastic support for veterans and the armed forces. He has also been critical of people who have been less than forthright about their Vietnam records. This month, he chided Sen. Richard Blumenthal, D-Conn., over misleading statements he made years ago about his own Vietnam record, calling him “Da Nang Dick” on Twitter.

Larry Braunstein’s daughters said that when he discussed Trump’s medical exemption, he often mentioned Weinstein, although it was unclear to them what role Weinstein may have played. He was close to the family, they said, and known as Uncle Manny.

The two men forged a close friendship after meeting in podiatry school in New York, from which they graduated in 1953. Weinstein was among the oldest students in the class, classmates said, and Braunstein was remembered for being among the smartest.

One possible explanation that has been raised over the years, the Braunstein sisters said, is that Weinstein had a connection to the draft, as some private practitioners did. In fact, multiple doctors would have been involved in the final determination.

Before people were inducted into the service, they underwent a physical exam overseen by military doctors, court records from that era show. Men could bring along documentation of medical concerns from private physicians. That information was presented at their exams and considered by a medical officer. Often, a civilian specialist working with the exam station would be asked to review the case and make a recommendation. A local draft board would finalize the man’s classification.

Weinstein practiced podiatry in Brooklyn’s Bath Beach neighborhood, maintaining an office near another Trump building, Shore Haven Apartments. In 1968, phone books show, Weinstein moved into an apartment in Westminster Hall, a Trump-owned building. He lived in that building for many years and later lived in another owned by the Trumps.

Weinstein had no children and never married, but some people who knew him were surprised by a possible Trump connection.

When Weinstein closed his practice in the late 1980s, he referred patients to a nearby podiatrist, Dr. Mark Schwartz. When contacted by The Times, Schwartz said he had never heard about a possible connection between Weinstein and the Trumps.

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