The Times’s investigation, based on a vast trove of confidential tax returns and financial records, reveals that Mr. 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 Mr. 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 Mr. 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 Internal Revenue Service, 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.