Iran’s leaders have perplexed several American presidents, perhaps none more than President Trump.

Mr. Trump considers himself a master of sizing up and exploiting his adversaries. But Iran’s leaders have baffled him.

“All they do is make me angry,” he told reporters last week. “They just make me angry.”

More frustration could be in store for the American president. Trump officials say a new deal to reopen the Strait of Hormuz may be near, but Iran continues to insist on collecting fees there that Mr. Trump has ruled unacceptable.

Mr. Trump’s inability to strike a favorable deal with Iran has not only cost him politically among war-weary voters, but continued strife around the crucial waterway has made for volatile oil prices at the expense of American consumers and the global economy.

Mr. Trump is hardly the first U.S. president who has struggled to read Iran. Each of his predecessors since the country’s 1979 Islamic Revolution has tried to decode Iran’s murky power structure and the mind-set of its political leadership.

Most have come up short; some have been burned badly. A rare exception is President Barack Obama, whose 2015 nuclear deal with Iran was a 20-month diplomatic effort. Mr. Obama considered it a major achievement, despite criticism from Republicans and some Democrats that Iran had lured him into a flawed deal by feigning moderation.

But Mr. Trump may be the president who “has least understood Iran’s leadership,” said Kenneth Pollack, a former C.I.A. analyst and National Security Council official who has studied Iran for decades.

“In many ways, Trump represents the epitome of America’s frustrations with Iran,” Mr. Pollack said. “He has been both the most desirous of making a deal with Iran, and the one who used the most force against it. He has found that neither got him what he wanted.”

Mr. Trump often speaks as though massive force will produce a quick deal.

After the war opened with a Feb. 28 airstrike that killed Iran’s supreme leader of nearly 40 years, Ali Khamenei, Mr. Trump called on Iran’s people to rise up and “take over” their government. He added that he would then “choose” Iran’s next leader.

More senior Iranian officials would also be killed in airstrikes. But no popular uprising emerged.

Iran’s hard-line government instead doubled down. In March, it named a new supreme leader, Mr. Khamenei’s son, Mojtaba Khamenei. U.S. officials consider him even more rigid — and more interested in acquiring a nuclear bomb — than his father.

Iran had simply ignored Mr. Trump’s public declaration that the younger Mr. Khamenei would be an “unacceptable” choice. And Mr. Khamenei did not respond to a public offer from Mr. Trump for an in-person meeting.

Even so, Mr. Trump retained hope for major change. “Iran can get themselves on a very good footing if they make a deal,” he said in late April. Iran, he said, could become “a legitimate country, not a country based on death and horror.”

Iranian officials say they can negotiate with Mr. Trump on matters such as the Strait of Hormuz and their nuclear program, even as they rule out many of his core demands.

But in a political system premised on hatred of the United States, none show interest in a larger reconciliation with the United States.

There are few signs that Mr. Trump’s months of dealing with Iran’s regime have given him a clearer understanding of it. Over the course of the conflict, Mr. Trump has constantly revised his assessment of Mr. Khamenei and his deputies. He has branded them “lunatics,” “reasonable,” “crazy bastards,” “very smart” and “stone cold crazy.”

Mr. Trump is to some degree experiencing frustrations that have bedeviled several of his predecessors who also hoped for grand bargains with Iran.

“For decades, American policymakers have consistently underappreciated the ideological commitment and deeply institutionalized structure of power in the Islamic Republic,” said Suzanne Maloney, vice president and director for foreign policy at the Brookings Institution in Washington.

One problem, Maloney noted, is how little direct exposure U.S. officials have had to top Iranian leaders. No U.S. official ever met Iran’s first supreme leader, Ayatollah Ruhollah Khomeini, or the late Mr. Khamenei, nor his son.

Ms. Maloney said that war appears to have strengthened the power of Iran’s religious and military leadership, leaving no “viable alternative power center.”

Iran’s semi-democratic political system, which features a president and parliament under the total religious authority of its supreme leader, has at times included high-level officials open to friendlier relations with the United States.

But past appeals by Washington to those figures have mostly ended in regret.

Dennis Ross, a veteran of Middle East diplomacy who served presidents of both parties, said that “the Iranians seem to understand us better than we understand them.