Searches of the DT for "oak", "pine", etc., will probably turn up a lot, but the trees probably won't be the real subjects of the songs, as they aren't in
"The oak and the ash and the bonnie ellum/birken tree
Are all a-growin' green in my own countree."
--"Home, Boys, Home"
The DT has a song called "Arbutus". A forum search for "arbutus" will turn up a different song I posted some months ago. It's as Celtic as your heart could desire, Cuilionn, but since the words are by the 19th-century poet Alfred P. Graves, it may not be what you're looking for.
There's also "The Cherry-Tree Carol"