I remember reading that the English had trouble getting enough arrows in their wars with France.
Egypt, a place with many fewer trees than England, fielded thousands of chariots, presumably carrying archers.
Where did all the arrows come from?
I'm no military historian...but I'm not totally sure i see the problem. Arrow shafts are not only made out of wood, and in fact most wood is kind of shitty, it's too heavy. Reed shafts are extremely well-attested in ancient literature, and Egypt isn't exactly lacking in reeds