Isn't history more fun when you know something about it?
This famous weapon was made of the native English Yew tree, and the act of drawing the longbow was known as "plucking the yew" (or "pluck yew").
Much to the bewilderment of the French, the English won a major upset and began mocking the French by waving their middle fingers at the defeated French, saying, "See, we can still pluck yew!"
Since 'pluck yew' is rather difficult to say, the difficult consonant cluster at the beginning has gradually changed to a labiodental fricative 'F', and thus the words often used in conjunction with the one-finger-salute!
It is also because of the pheasant feathers on the arrows used with the longbow that the symbolic gesture is known as "giving the bird."
And yew thought yew knew everything!