The expert on this is Luther S. Harris, the author of "Around
Washington Square." While I don't think he goes into it in depth in
that text, he's done a lot of research to definitively prove that a)
the city of New York didn't use trees to hang people; and b) that the
elm in question wasn't even in the boundaries of the old potters field
at the time and that it quite unlikely that the city would have hanged
criminals on private property when they had a graveyard and gallows
quite close by.

(Perhaps Mr. Harris is on the list and can answer you in further detail.)

My own research into why the elm has become "the hanging elm" over the
years has led to the likely-but-unverifiable conclusion that it is
simply because people like something they can point to and say,
"That's where they hanged people when this was a graveyard." Trying to
conjure up pictures of a long-extinct gallows is difficult; a tree
with a nice plaque that says "Hainging Elm" is easy.

Regards,
James

-- 
James Nevius
http://www.walknyc.com