The flowers are small and white.

Thanks, I always thought that Privet was a bush, not a tree but I just looked them up, it certainly appears to be the case.

It's unfortunate that they seem to prefer moist soil as they are in a heavily shaded dry area. That being said, they are not being watered and are surviving so maybe they have developed some drought tolerance in shaded areas?

They become very tough and drought tolerant when thoroughly established, yes.

Shade also reduces water need, though it doesn't do much for their looks. They may be remnants of a hedge that was once in full sun, or they may have been volunteer seedlings that sprouted the seed being deposited by birds. Birds like the berries.

A lot of plants we treat as shrubs are actually small trees in their natural habitat. There are a few Ligustrum in the neighborhood that are 30' tall.

Yeah, looks just like a Privet. My back neighbor has a few right on the fence line. I don't think they water them at all and they look great. If they do get water it is probably from my side of the fence :) In my area (Livermore), Cedar Waxwings and Robins gather in large groups to eat the berries, which is really cool.

Some of ours are 6 ft. tall, some (with a bit more sun exposure) are 20+ ft. tall. I think that they were originally a hedge as they are lined up in a somewhat linear fashion behind the back fence. We are restoring badly neglected (overgrown) property, everything is an adventure and new knowledge. This site is great.

I've seen privets in the Houston, TX area of monster size.