The weeping willow motif is very common in early folk art as a sign of mourning. I recently saw a mourning painting with a huge, overpowering willow drooping over the mourners and the grave- surely willows in music have the same significance.
Let's see if I can remember this old English song

A poor soul sat sighing
All under a tree
Singing all a green willow,
O, willow, willow, willow willow
My garland shall be.