An emblem of being forsaken. “All round my hat I wear a green willow.” So Shakespeare:
“I offered him my company to a willow-tree to make him a garland, as being forsaken.” (Much Ado About Nothing, ii. 1.) The very term weeping willow will suffice to account for its emblematical character.
Source: Dictionary of Phrase and Fable, E. Cobham Brewer, 1894