DigiTrad contains about 30 songs that contain the phrase "willow tree". "Willow Tree" is not the title of any of them. See the "Mudcat FAQ" - the first thread on the Forum page - for information about how to search DigiTrad.

This song, and the next one, are probably not the song you want, but since I ran across them while searching, and they don't seem to already be anywhere in the forum or DigiTrad, I may as well add them here:

Copied from http://www.btinternet.com/~radical/thefolkmag/hamer/hamer13.htm

THE WILLOW TREE
Sung by May Bradley, Shropshire

As I passed by a willow tree, willow tree,
That willow leaf blew down on me.
I picked it up, it would not break.
I passed my love, he would not speak.

Oh, speak, young man, and don't be shy, be shy,
I'm not a girl can pass you by,
For friends we met and friends we'll part,
Just take my hand but not my heart.

I wish your bosom was of glass, of glass,
That I could view it through and through,
Just view those secrets of your heart,
If I love one I can't love two.

Then give me back to the one I love, I love,
Oh, give, oh give him back to me,
If I only had that one I love,
How happy, happy should I be.

My love he is a sailor boy, sailor boy,
He sails the ocean through and through,
And when he gets so far away,
He hardly thinks no more of me.

Now give me back to the one I love, I love,
Oh give, oh give him back to me,
If I only had that one I love,
How happy, happy should I be.