While Mr. Trump is skipping a summit meeting of Southeast Asian leaders in the Philippines to head home a day earlier, it will still be the longest tour of Asia by any president since the elder George Bush in late 1991 — a trip that ended in Tokyo when Mr. Bush, tired and ill with the flu, collapsed after vomiting on the Japanese prime minister.

In Vietnam, his aides said, Mr. Trump will articulate a new policy for Asia built on the concept of a “free and open Indo-Pacific” region. The idea, they acknowledge, originated with the Japanese, who have been urging the United States to bond with three other maritime democracies — Japan, Australia, and India — to contain a rising China.

“This trip is a great opportunity to demonstrate America’s, and the Trump administration’s, commitment to the Indo-Pacific,” said the national security adviser, Lt. Gen. H. R. McMaster, using what seemed likely to become a catchphrase for the Trump administration, much as the “Asia pivot” was for the Obama administration.

Japanese officials planted the Indo-Pacific idea with two American counterparts: Brian H. Hook, the State Department’s policy planning director, and Matthew Pottinger, the Asia director in the National Security Council. But it dates further back, to a 19th-century American naval officer and historian, Alfred Thayer Mahan, whose writings about maritime power have long been studied in Japan but who has only recently drawn attention in the White House.

The problem is, it is unclear that Mr. Trump is bringing any initiatives to Asia to further that vision. He withdrew the United States from the most obvious regional project, the Trans-Pacific Partnership. He is pushing bilateral trade deals to replace that accord, but Japan and other Asian countries are reluctant to open negotiations, while South Korea is balking at Mr. Trump’s demand to renegotiate its existing trade agreement.