Pic. The berries are maybe slightly larger than a green pea. Central South Carolina, and these trees grow like weeds.
It looks like White Ash.
I’m thinking Black Cherry (Prunus serotina). It has alternate leaves, the Ash has opposite leaves.
They look kinda like Chokecherries (Prunus virginiana) although the Carolinas are a little far south for them. Could also be Chokeberries (Aronia) which does apparently grow there, although the leaves don’t look quite right.
The Larch
Crush the leaves or peel some bark and give it the sniff test. If it smells like cherry, it probably is. Note that not all cherries in the southern US are edible; they have too much prussic acid.
Some variety of tupelo?
No, clearly not. Tupelo fruit may look similar, if you look only at it, but tupulus flower and fruit only at the end of the branch.
The OP’s tree has the fruit back up before a bunch of leaves.
Also the leaves have the centre vein, but no more veins visible.
Also the leaves have that dimply or wavy edge, and the same texture/colour.
So its Prunus… So along with cherry like fruit, dark coloured, thats four distinctive characteristics of Prunus…
The leaves do look different at different ages of tree, and tree people generally want the bark to identify the tree .
But prunus is growing well in the Carolinas , and as a tree ? Its Black Cherry (Prunus serotina)
I thought the leaves did look similar, that’s why I thought it was a White Ash. Sorry about that.
The fruit however, looks nothing like White Ash. Brain lapse.
The leaves are completely different. Not to pick on you, but perhaps some fundamentals of plant identification are in order. White ash has opposite compound leaves (a very unusual combination among North American trees), while the tree in the photos has alternate simple leaves (very common). Even the shapes of the leaflets/leaves are rather different, ash having somewhat ovate leaflets with a broad base, while the leaves in the photos are more elliptical and narrow towards the base. Since we have lots of experts here, it’s better not to jump in with your first impression.
Those fruit aren’t on a raceme. Black cherries bear fruit on racemes. So do choke cherries and pin cherries. Is there more than one seed in each fruit? OP said they’re berries, and cherries aren’t berries, they’re drupes.
Was the OP correct in calling them berries though?
The… Larch
He could be using the term in the loose sense, for any small fleshy fruit, and not the strict botanical one.
However, both P. serotina and P. virginiana have serrate leaf margins. The photo is pretty crappy, but it doesn’t look like they have serrate margins to me.
I think we need better photos and more information from the OP.
Yeah. Botanical presciptivists. Can’t live with 'em, and it’s against the rules to make fun of
'em. 
I’m going with some type of Tupelo.
How about hackberry? Alternate leaves, little purple berries arranged that way. Leaves are serrated, but there’s a lot of species. They commonly grow as “weed trees” also.