Movie about a American GI falling in love with a Japanese girl during ww11. They meet under a tree.

I'm not sure of this, as he is a war correspondent, not a soldier:

Love Is a Many-Splendored Thing (1955)
From Wikipedia:
The widowed Eurasian doctor Han Suyin (Jennifer Jones) falls in love with the married-but-separated American correspondent Mark Elliott (William Holden) in Hong Kong, during the period of China's Civil War in the late 1940s. Although they briefly find happiness together, she is ostracized by the greater Chinese community. After losing her position at the hospital, Suyin and her adopted daughter go to live with a friend while Mark is on an assignment during the Korean War. They write to each other constantly.

Here's the theme song with clips from the movie, including them meeting under a tree: https://www.youtube.com/watch?v=DWYLK4rtH-w



(edited for correction)