There is a mature, 25' tall palm about 8 feet away, and I suspect this palm came from it's seed. Here are some pictures of the mature palm. It's tall and skinny, single stem.

The base of the palm.

It has clusters of flowers and seeds up high, kind of hard to see with the sun and the glare.

Some flowers dropped from above and they are small (dime size) and white.

It also drop some red berry like fruit (seeds).

Is this the parent plant? After looking at some online pictures I am thinking the parent plant is either Christmas Palm or Carpentaria Palm?

Are the two smaller palms (both now about 5-6 feet tall) the same palm?

Thank you, so the mature palm is the same species as the two smaller new palm nearby? They are both the same?

It's common name is Solitaire Palm but also some places call it Alexander Palm?

I'm thinking Adonidia merrillii or Christmas palm, due to the bright red fruits and the smooth green trunk on the young one.

http://gardeningsolutions.ifas.ufl.edu/plants/trees-and-shrubs/palms-and-cycads/christmas-palm.html

Between Adonidia, Ptychosperma and Carpentaria I'd discount Carpentaria on the small plants. The fronds don't arch the same way. And I don't see the first few leaflets with long thin 'threads' hanging down from the tips as you see with Adonidia. They're like long thin strips that are much longer than the main leaflets themselves.

With the adult palm the seeds aren't Adonidia, they're a bit like Carpentaria but more like Ptychosperma. The fronds on the adult (from a distance) aren't Carpentaria, but they're similar to both Adonidia and Ptychosperma.

Edited to comment on growth speed, Ive found Carpentaria to be fastest, Ptychosperma not so fast, and Adonidia fairly slow.

Definitely Ptychosperma elegans, the fruit of Adonidia is much larger.

After looking up the few possible matches, I am inclined to think it is Ptychosperma elegans as well.

The only thing that is still confusing me is I went to look at palms at a nursery last weekend, and I saw some Ptychosperma elegans for sale. However these Ptychosperma elegans are not single stem palms, they have three in each pot. My palm is single stem. Does it mean they planted three in each pot for the look, or does it mean it's the same root ball with a cluster of three stems?

Some Ptychosperma species are clumping palms, but not elegans.

By the way, the label shown in your photo is wrong. P. elegans is called Solitaire Palm but not Alexander Palm which are actually Archontophoenix alexandrae, a totally different genus of palm.

Ptychosperma elegans is a single stemmed palm. But they are often potted as multiples, same as Adonidia and Phoenix roebelenii.

The larger palm is a Ptychosperma elegans and smaller one is a Adonidia Merrillii