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UNITED NATIONS — Diplomats at the United Nations have by now learned to expect the unexpected from President Trump, who threatened to annihilate North Korea and “crush loser terrorists” during his debut speech at the General Assembly a year ago.

But at this year’s convergence of world leaders in New York, which begins in earnest on Tuesday, Mr. Trump will command an additional stage. Aside from his speech, he will lead a meeting of the 15-member Security Council, the most powerful body of the United Nations.

There are plenty of pressing issues for world leaders to discuss — wars in Yemen and Syria that have killed hundreds of thousands of people; negotiations over North Korea’s nuclear program; and the fate of the Iran nuclear accord, which Mr. Trump repudiated but European leaders want to preserve.

Underlying Mr. Trump’s appearance is an increased sense that the United States, which helped found the United Nations after World War II, is isolating itself. His administration has moved to cut the United Nations budget and renounced participation in some of the organization’s most prominent agreements and agencies.