By DAVID BARSTOW, SUSANNE CRAIG and RUSS BUETTNER

President Donald Trump participated in dubious tax schemes during the 1990s, including instances of outright fraud, that greatly increased the fortune he received from his parents, an investigation by The New York Times has found.

Trump won the presidency proclaiming himself a self-made billionaire, and he has long insisted that his father, legendary New York City builder Fred C. Trump, provided almost no financial help.

But The Times’ investigation, based on a vast trove of confidential tax returns and financial records, reveals that Trump received the equivalent today of at least $413 million from his father’s real estate empire, starting when he was a toddler and continuing to this day.

Much of this money came to Trump because he helped his parents dodge taxes. He and his siblings set up a sham corporation to disguise millions of dollars in gifts from their parents, records and interviews show. Records indicate that Trump helped his father take improper tax deductions worth millions more. He also helped formulate a strategy to undervalue his parents’ real estate holdings by hundreds of millions of dollars on tax returns, sharply reducing the tax bill when those properties were transferred to him and his siblings.

These maneuvers met with little resistance from the IRS, The Times found. The president’s parents, Fred and Mary Trump, transferred well over $1 billion in wealth to their children, which could have produced a tax bill of at least $550 million under the 55 percent tax rate then imposed on gifts and inheritances.

The Trumps paid a total of $52.2 million, or about 5 percent, tax records show.

The president declined repeated requests over several weeks to comment for this article. But a lawyer for Trump, Charles J. Harder, provided a written statement on Monday, one day after The Times sent a detailed description of its findings. “The New York Times’ allegations of fraud and tax evasion are 100 percent false, and highly defamatory,” Harder said. “There was no fraud or tax evasion by anyone. The facts upon which The Times bases its false allegations are extremely inaccurate.”

Harder sought to distance Trump from the tax strategies used by his family, saying the president had delegated those tasks to relatives and tax professionals. “President Trump had virtually no involvement whatsoever with these matters,” he said. “The affairs were handled by other Trump family members who were not experts themselves and therefore relied entirely upon the aforementioned licensed professionals to ensure full compliance with the law.”

The president’s brother, Robert Trump, issued a statement on behalf of the Trump family:

“Our dear father, Fred C. Trump, passed away in June 1999. Our beloved mother, Mary Anne Trump, passed away in August 2000. All appropriate gift and estate tax returns were filed, and the required taxes were paid. Our father’s estate was closed in 2001 by both the Internal Revenue Service and the New York state tax authorities, and our mother’s estate was closed in 2004. Our family has no other comment on these matters that happened some 20 years ago, and would appreciate your respecting the privacy of our deceased parents, may God rest their souls.”

The Times’ findings raise new questions about Donald Trump’s refusal to release his income tax returns, breaking with decades of practice by past presidents. According to tax experts, it is unlikely that Trump would be vulnerable to criminal prosecution for helping his parents evade taxes because the acts happened too long ago and are past the statute of limitations. There is no time limit, however, on civil fines for tax fraud.

The findings are based on interviews with Fred Trump’s former employees and advisers and more than 100,000 pages of documents describing the inner workings and immense profitability of his empire. They include documents culled from public sources — mortgages and deeds, probate records, financial disclosure reports, regulatory records and civil court files.

The investigation also draws on tens of thousands of pages of confidential records — bank statements, financial audits, accounting ledgers, cash disbursement reports, invoices and canceled checks. Most notably, the documents include more than 200 tax returns from Fred Trump, his companies and various Trump partnerships and trusts. While the records do not include the president’s personal tax returns and reveal little about his recent business dealings at home and abroad, dozens of corporate, partnership and trust tax returns offer the first public accounting of the income he received for decades from various family enterprises.

What emerges from this body of evidence is a financial biography of the 45th president fundamentally at odds with the story Trump has sold in his books, his TV shows and his political life. In Trump’s version of how he got rich, he was the master dealmaker who broke free of his father’s “tiny” outer-borough operation and parlayed a single $1 million loan from his father (“I had to pay him back with interest!”) into a $10 billion empire that would slap the Trump name on hotels, high-rises, casinos, airlines and golf courses the world over. In Trump’s version, it was always his guts and gumption that overcame setbacks. Fred Trump was simply a cheerleader.

“I built what I built myself,” Donald Trump has said, a narrative that was long amplified by often-credulous coverage from news organizations, including The Times.

Certainly a handful of journalists and biographers, notably Wayne Barrett, Gwenda Blair, David Cay Johnston and Timothy L. O’Brien, have challenged this story, especially the claim of being worth $10 billion. They described how Trump piggybacked off his father’s banking connections to gain a foothold in Manhattan real estate. They poked holes in his go-to talking point about the $1 million loan, citing evidence that he actually got $14 million. They told how Fred Trump once helped his son make a bond payment on an Atlantic City casino by buying $3.5 million in casino chips.

But The Times’ investigation of the Trump family’s finances is unprecedented in scope and precision, offering the first comprehensive look at the inherited fortune and tax dodges that guaranteed Donald Trump a gilded life. The reporting makes clear that in every era of Trump’s life, his finances were deeply intertwined with, and dependent on, his father’s wealth.

By age 3, Trump was earning $200,000 a year in today’s dollars from his father’s empire. He was a millionaire by age 8. By the time he was 17, his father had given him part ownership of a 52-unit apartment building. Soon after Trump graduated from college, he was receiving the equivalent of $1 million a year from his father. The money increased with the years, to more than $5 million annually in his 40s and 50s.

Fred Trump’s real estate empire was not just scores of apartment buildings. It was also a mountain of cash, tens of millions of dollars in profits building up inside his businesses, banking records show. In one six-year span, from 1988 through 1993, Fred Trump reported $109.7 million in total income, now equivalent to $210.7 million. It was not unusual for tens of millions in Treasury bills and certificates of deposit to flow through his personal bank accounts each month.

Fred Trump was relentless and creative in finding ways to channel this wealth to his children. He made Donald Trump not just his salaried employee but also his property manager, landlord, banker and consultant. He gave him loan after loan, many never repaid. He provided money for his car, money for his employees, money to buy stocks, money for his first Manhattan offices and money to renovate those offices. He gave him three trust funds. He gave him shares in multiple partnerships. He gave him $10,000 Christmas checks. He gave him laundry revenue from his buildings.

Much of his giving was structured to sidestep gift and inheritance taxes using methods tax experts described to The Times as improper or possibly illegal. Although Fred Trump became wealthy with help from federal housing subsidies, he insisted that it was manifestly unfair for the government to tax his fortune as it passed to his children. When he was in his 80s and beginning to slide into dementia, evading gift and estate taxes became a family affair, with Donald Trump playing a crucial role, interviews and newly obtained documents show.

The line between legal tax avoidance and illegal tax evasion is often murky, and it is constantly being stretched by inventive tax lawyers. There is no shortage of clever tax avoidance tricks that have been blessed by either the courts or the IRS itself. The richest Americans rarely if ever pay anything close to full freight.

But tax experts briefed on The Times’ findings said the Trumps appeared to have done more than exploit legal loopholes. They said the conduct described here represented a pattern of deception and obfuscation, particularly about the value of Fred Trump’s real estate, that repeatedly prevented the IRS from taxing large transfers of wealth to his children.

“The theme I see here through all of this is valuations: They play around with valuations in extreme ways,” said Lee-Ford Tritt, a University of Florida law professor and a leading expert in gift and estate tax law. “There are dramatic fluctuations depending on their purpose.”

The manipulation of values to evade taxes was central to one of the most important financial events in Donald Trump’s life. In an episode never before revealed, Trump and his siblings gained ownership of most of their father’s empire on Nov. 22, 1997, a year and a half before Fred Trump’s death. Critical to the complex transaction was the value put on the real estate. The lower its value, the lower the gift taxes. The Trumps dodged hundreds of millions in gift taxes by submitting tax returns that grossly undervalued the properties, claiming they were worth just $41.4 million.

The same set of buildings would be sold off during the next decade for more than 16 times that amount.

The most overt fraud was All County Building Supply & Maintenance, a company formed by the Trump family in 1992. All County’s ostensible purpose was to be the purchasing agent for Fred Trump’s buildings, buying everything from boilers to cleaning supplies. It did no such thing, records and interviews show. Instead All County siphoned millions of dollars from Fred Trump’s empire by simply marking up purchases already made by his employees. Those millions, effectively untaxed gifts, then flowed to All County’s owners — Donald Trump, his siblings and a cousin. Fred Trump then used the padded All County receipts to justify bigger rent increases for thousands of tenants.

After this article was published on Tuesday, a spokesman for the New York state Department of Taxation and Finance said the agency was “reviewing the allegations” and “vigorously pursuing all appropriate areas of investigation.”

All told, The Times documented 295 streams of revenue that Fred Trump created over five decades to enrich his son. In most cases his four other children benefited equally. But over time, as Donald Trump careened from one financial disaster to the next, his father found ways to give him substantially more money, records show. Even so, in 1990, according to previously secret depositions, Trump tried to have his father’s will rewritten in a way that Fred Trump, alarmed and angered, feared could result in his empire being used to bail out his son’s failing businesses.

Of course, the story of how Donald Trump got rich cannot be reduced to handouts from his father. Before he became president, his singular achievement was building the brand of Donald Trump, Self-Made Billionaire, a brand so potent it generated hundreds of millions of dollars in revenue through TV shows, books and licensing deals.

Constructing that image required more than Fred Trump’s money. Just as important were his son’s preternatural marketing skills and always-be-closing competitive hustle. While Fred Trump helped finance the accouterments of wealth, Donald Trump, master self-promoter, spun them into a seductive narrative. Fred Trump’s money, for example, helped build Trump Tower, the talisman of privilege that established his son as a major player in New York. But Donald Trump recognized and exploited the iconic power of Trump Tower as a primary stage for both “The Apprentice” and his presidential campaign.

The biggest payday he ever got from his father came long after Fred Trump’s death. It happened quietly, without the usual Trumpian news conference, on May 4, 2004, when Trump and his siblings sold off the empire their father had spent 70 years assembling with the dream that it would never leave his family.

Donald Trump’s cut: $177.3 million, or $236.2 million in today’s dollars.