Donald J. Trump built a business empire and won the presidency proclaiming himself a self-made billionaire, and he has long insisted that his father, the legendary New York City builder Fred C. Trump, provided almost no financial help. “I built what I built myself,” the president has repeatedly said.

But an investigation by The New York Times has revealed that Donald Trump received the equivalent today of at least $413 million from his father’s real estate empire. What’s more, much of this money came to Mr. Trump through dubious tax schemes he participated in during the 1990s, including instances of outright fraud, The Times found.

In all, the president’s parents 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 on gifts and inheritances that was in place at the time. Helped by a variety of tax dodges, the Trumps paid $52.2 million, or about 5 percent, tax returns show.

The president declined requests over several weeks to comment for this article.

A lawyer for Mr. Trump, Charles J. Harder, provided a written statement. “There was no fraud or tax evasion by anyone. The facts upon which The Times bases its false allegations are extremely inaccurate,” he said. “President Trump had virtually no involvement whatsoever with these matters,” he continued, saying the president had delegated those tasks to relatives and tax professionals. “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.”