Thanks, Sandman. At https://afolksongaweek.wordpress.com/2016/09/02/week-261-the-hawk-and-the-crow/ it's called The Hawk and the Crow, but it says

"I learned this from one of my absolute favourite singers, Kevin Mitchell, via his 1977 Topic LP Free and Easy. Kevin calls it ‘Two Strings on a Bow’, and he learned it from Anne Brolly of Dungiven, County Derry. The LP notes say

American singers call this song The Bird’s Courtship or The Leather Winged Bat."

So evidently this is a song of many titles.