till Birnam wood do come to . . . Yamama?

I was reading Juvaini's History of the World Conqueror, and when I reached his chapter (Chapter XVI) on the Mongol march to Bokhara I had to do a double-take. Juvaini describes a detatchment of Mongols disguising themselves as merchants to take a town, and to add colour he compares it with the following folk story:

"It is the story of Zarqa of Yamama. She had built a lofty castle, and her keenness of sight was such that if an enemy attempted to attack her she would descry his army at a distance of several stages and would prepare and make ready to repel him and drive him off. And so her enemies achieved nought but frustration and there remained no strategem which they had not tried. Finally one of them commanded that trees should be cut down with their branches and that each horseman should hold a tree in front of him. Thereupon Zarqa exclaimed: 'I see a strange sight: the likeness of a forest is moving towards us.' Her people said: 'The keenness of thy sight hath suffered some hurt, else how should trees move?' They neglected to watch or take precautions; and on the third day their foes arrived, and overcame them, and took Zarqa prisoner, and slew her."

This seems too close for coincidence to the attack on Dunsinane in the final act of Macbeth:

(Act 5, Scene 4)
Siward: What wood is this before us?
Menteith: The wood of Birnam.
Malcolm: Let every soldier hew him down a bough,
And bear't before him; thereby shall we shadow
The numbers of our host, and make discovery
Err in report of us.

So just how old is this story, and where does it come from? I do wonder how an old Arabic folk tale would have found its way to Elizabethan England, to be picked up by a London playwright.