Songs "about" trees is a tall order, since, well, they aren't the most active little buggers unless they fall on your house, so mostly they appear either symbolically or as sites of more cogent happenings.

"'Tis Autumn" by Bob Durough: "The trees say they're tired, they've borne too much fruit..."
"Lemon Tree": Trini Lopez standard
"Tie a Yellow Ribbon Round the Old Oak Tree"
"The Liberty Tree"
"Barbara Allen": In the end, aren't she and her would-be lover turned into trees (or bushes)?
"O Tannenbaum" / "O Christmas Tree"
"Indian Love Call": "When I'm calling 'yewwwww'" (couldn't resist)

A couple Russian ones:
"Stanu ja cherjomukhoj" (I'll become a cherry tree)
"Vo polje berjozenka stojala" (A birch tree stood in the field)

Don't suppose you'd want logging songs.