It was also a demonstration, however coincidental, of the political shadow boxing that has found an unlikely arena in Europe, the new center of the contest between liberal democracy and far-right populism.
While Mr. Obama is the leader Europe prefers, Mr. Trump’s sudden ascendance has been seen as a challenge to America’s commitment to Europe, both its unity and its security, as well as the values that underpin the Western alliance.
The impression was underscored once again on Thursday when Mr. Trump demurred from explicitly endorsing America’s commitment to NATO’s principle of collective defense.
Neither president has remained aloof from Europe’s politics as the stakes have mounted this year with critical elections that have so far beaten back the far-right populism that helped thrust Mr. Trump to power last year.
Each man has, in fact, made his preferences clear at important moments in a kind of political proxy war. Mr. Obama, who remains wildly popular in Europe, was not shy about weighing in on France’s presidential race and endorsing the centrist reformer Emmanuel Macron, the winner.
Mr. Trump, on the other hand, lauded Mr. Macron’s far-right opponent, Marine Le Pen, and posted a message on Twitter saying a terror attack in Paris in April would “have a big effect on presidential election!” Ultimately it did not.
For European leaders like Ms. Merkel, striking a balance between coaxing Mr. Trump into a deeper understanding with America’s traditional allies, while remaining true to their own political principles, is proving to be tricky.
German government officials say Ms. Merkel telephoned Mr. Trump when it became clear she would meet both presidents on the same day, to dispel any impression of a slight.
But the coincidence of scheduling — Mr. Obama’s invitation was issued a year ago, though accepted only last month — nonetheless presented Ms. Merkel with an opportunity for her to demonstrate that both sides need each other, and to show voters at home that she is a world leader as she campaigns for a fourth term.
“It is wonderful timing for her, a combination of good luck and good strategizing,” said Jan Techau of the Richard Holbrooke Forum at the American Academy in Berlin.
She was with Mr. Obama, “the good American who everyone is already missing,” and then with President Trump, “the other America which needs to be dealt with. And that is what is so crucial — of course she needs the relationship with Trump, but she can relativize that with pictures with Obama at the church meeting,” Mr. Techau added.
Yet, in Brussels, there were no evident breakthroughs.
As for Mr. Obama, usually trips by ex-leaders generate little public interest and consist of collecting obscure awards, like the media prize Mr. Obama was due to accept in the German spa town of Baden-Baden later on Thursday.
But while Mr. Obama has generally avoided making overtly political statements during his travels, his every movement, gesture and word have become objects of scrutiny at a highly politicized time.
Mr. Obama took his first step back onto the world stage earlier this month, at a food and technology conference in Milan, where he sprinkled his political stardust on Matteo Renzi, the center-left former Italian prime minister who is hoping for a comeback.
The themes and settings of this week scarcely spelled neutrality, or reserve, analysts noted. “The entire week is more about symbolism than it is about substance,” Mr. Techau said. “It is state theater at the highest level.”
Mr. Obama did not mention Mr. Trump once during his 90-minute appearance in Berlin. But he did take some veiled swipes, noting, for instance, that when dealing with migration, “we can’t hide behind a wall,” alluding to Mr. Trump’s plan to build a wall along the Mexican border.
In Brussels, Ms. Merkel, was similarly discreet as she unveiled a piece of the Berlin Wall, whose fall in November 1989 marked NATO’s triumph in the Cold War against the Soviets.
“To find convincing answers for the future,” she said, “it is good to know what we achieved in the past.”
Mr. Trump, the New Yorker, presented a large chunk of the North Tower of the World Trade Center where the first hijacked plane made impact on Sept. 11, 2001, leading NATO allies for the first time to invoke the collective defense clause, Article V, which European leaders were hoping Mr. Trump would endorse.
Instead, Mr. Trump wasted no time in reminding Europeans that most of them are not paying their way in defense, and that this is “not fair” to the American taxpayer.
While the atmosphere in Brussels was tense, in Berlin Germans and foreigners exulted in the chance to see and hear Mr. Obama live.
Austin Joseph, 27, a native of Atlanta, said he left the United States two days after Mr. Trump’s election and swiftly settled in Berlin. “They talked to each other with decency and respect,” he said after Mr. Obama’s appearance with Ms. Merkel. “That is what we need more of nowadays.”
The very different sentiments evoked by Mr. Trump are equally clear.
“Donald Trump is not capable of being President of the U.S.A.,” wrote Klaus Brinkbäumer, the editor of Der Spiegel, in an extended editorial in the current issue.
The 45th president is neither intellectually nor morally equipped for the job, he wrote. “Trump must be removed from the White House. Fast. He is a danger for the world.”
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