Thank you, Monique. The song is known to almost everyone who ever lived in France; I remember it well myself, though I was already at schoolboy age when I moved there.

The words and the melody are such as may well have been improvised by a 19th century nanny to keep the kids busy. It must have happened in a big city, since any country dweller would know that deer (and hares) just run away when hunted, whereas rabbits live in self-built burrows.

"Cerf" and "fenêtre" do not rhyme by any standard, which makes me believe that the French first verse may indeed be the earliest. The second verse and most of the translations sound as if designed by expert poets, at a time when the first verse was already popular.

But we know that there were poets who liked to mimick "real folklore", so that I am by no means sure. However, there must be some evidence from earlier than 1937.