Could be a Yaupon (Ilex vomitoria). It will have berries provided it's

a) a female and
b) there's a male within range to pollinate it.

You don't say where you live. That would help with the pollination issue and the likelihood that it's a Yaupon.

We live in Northwest Louisiana.

It looks like a yaupon to me. In Louisiana, there should be plenty of male pollinators nearby.

Have you ever seen the tiny light green/yellow flowers on the branches? The other (very likely) possibility is that you have the popular cultivar "nana", a compact variety widely used for foundation plantings. It is a dwarf female, but rarely flowers or fruits.

I wouldn't worry about yellow leaves or a few drying branches. This is normal for a yaupon, especially for one in bright sun. As long you're getting plenty of new growth each year, it should be fine. You can even chop off the canopy if you don't like the look, and it will grow back from the base into a more mounded shape.

Thank you greentiger87. Two out of the three have new growth. One of them is struggling. It is the one with the least amount of sun.

Yes, in LA you probably have native yaupons in the woods.

Thank you for your info. This is the struggling Yaupon. It is still loosing leaves. So I should not worry about all of the dead on it?

Dieback is common as the the dwarf varieties get too large. I would cut it back hard and toss some organic fertilizer around. If it doesn't rebound within the year, then replace it.