choosing between webbing or rope isn't going to help or cause the hammock to lean to one side. if you had the support rope or webbing on one side of the hammock, i suppose it could make some difference, but i've never seen that to be the case.
like blackbishiop said, it's most likely the hammock material or getting in & out one side all the time.
but i use mine at home exclusively & get in & out the same side... sometimes for months at a time w/ the same hammock.
my guess would be the fabric doesn't have very good memory (like me<g>) or it wasn't gathered correctly at the ends when it was put together.
as for using rope on the trees, i will share my opinion on that.
i've seen first hand what rope can do to a tree. webbing can do damage too, but not nearly as much or as likely.
it may not kill the tree, but it sure don't help it.
and if we want to continue to pass along a good name for hammock camping, it's in our best interest (as well as the trees) to encourage using webbing around the trees. ...tim
don`t leave the CREATOR out of the creation!