A senior White House official told reporters that the two leaders did not keep score, but enjoyed each other's company.
Ballistic missile tests by North Korea and its sixth and largest nuclear test, in defiance of UN Security Council resolutions, have exacerbated the most critical global challenge of Mr. Trump's presidency.
He added that North Korea was a "big problem for our country and for the world, and we want to get it solved" but had kind words for the people in the hermit state.
He presented the USA with a hat embroidered with the words "Donald and Shinzo: Make Alliance Even Greater" - a play on Mr Trump's 2016 election campaign theme.
When Trump hosted Abe in Palm Beach earlier this year, they played at one of Trump's Florida golf courses. The two passed up the region's Kobe beef in favour of the American version, which is favoured by Trump, a picky eater. "We like each other and our countries like each other", Trump said before dinner with Abe, who for this meal did show Trump traditional cuisine with a teppanyaki dinner. For that outing, Trump brought along pro golfer Ernie Els, so this time Abe matched him by bringing along Matsuyama, whom Trump described on the plane ride to Asia as "probably the greatest player in the history of Japan".
The North's state-run KCNA news agency said in a commentary that the USA should be disabused of the "absurd idea" that Pyongyang would succumb to global sanctions and give up its nuclear weapons, adding that it is in "the final stage for completing nuclear deterrence".
"What's inflammatory is the North Korean regime and what they're doing to threaten the world", McMaster said.More news: UK Service Sector Expands Most In 6 Months On Strong Orders
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Both leaders are also expected to hold a joint press conference later during this trip.
He warned that Washington and its allies are prepared to defend freedom and that "no dictator" should underestimate U.S. resolve.
Prime Minister Shinzo Abe has gone out of his way to cultivate close relations with Trump.
Trump told Fox News this week that it was possible he would meet Putin during his Asia trip.
"Together with our allies, America's warriors are prepared to defend our nation using the full range of our unmatched capabilities". Mr. Abe and Mr. Trump will meet families of Japanese citizens abducted by North Korea.
Relations between Moscow and Washington have soured further since Putin and Trump first met at a G20 summit in Hamburg in July when they discussed allegations of Russian meddling in the U.S. election, but agreed to focus on better ties.
Mr Trump has rattled some allies with his vow to "totally destroy" North Korea if it threatens the USA and his dismissal of North Korean leader Kim Jong Un as a "rocket man" on a suicide mission. He has dropped that antagonist language nearly entirely since the election, but tensions remain.