There are weeping cherries. Try looking up pictures of them to compare. I included a link to a page on them below.

FataMorgana

Here is a link that might be useful: Weeping Cherry (Prunus subhirtella 'Pendula')

Look up Pyrus salicifolia 'Pendula'

Resin

The difference in leaves between the weeper and the suckers are most likely due to it being grafted. The suckers are coming from the rootstock.

tj

You would probably get a more attractive specimen without the heavy pruning. Certainly less bother.

Salix purpurea 'Nana' or 'Pendula' grafted onto a standard. The growth that looks non-typical is a sprout and is best pruned off. It's on some kind of willow understock; I don't know what.

As far as whether or not it's a cherry, highly unlikely. Grafts are usually done on closely related species, and almost always within families. There are certain plants called chimeras, but they are rare, and all I can think of are still within a family association.

Ding!

Ding, ding, ding! You guys/girls are great! Mystery solved! I do believe I have a Salix purpurea 'Pendula'.
I looked on the Waverly Farms website & mine looks just like that except the lower branches have been pruned more. They graft theirs on a 5' standard.

I also checked my Michael Dirr tree book and it says the 'Pendula" is usually grafted on a standard.

Thank you everyone! I knew I could count on you. It's a beautiful little tree, the branches sway in the wind and the limbs go to the ground in the summer - it makes a nice screen - I highly recommend.

Plenty of them are also sold on their own roots. I've got one such cascading over a wall. Anyway, since it's a willow there probably will be a need for occasionally pruning it back hard to clear out old stems - I just wouldn't automatically whack it back every year because it was cut back before. Think in terms of periodic rejuvenation, where it is pruned back to stubs or short lengths of stem at the end of winter each time it starts to look shabby or overgrown. Don't snip at it much otherwise, pruning stems back only part way will tend not to produce a satisfactory appearance, unless done only to a limited extent.