This spring I didn't notice any pollen shed from the two seedling persimmons, but the Morris Burton appears to have set about a dozen fruit.

As far as I know, there are no other persimmon in this area.

Is it possible for persimmon to exhibit apomixis? In other words, can a female American persimmon sometimes be self-fruitful? Anyone heard of this?

Maybe I just missed the pollen shed on one of the seedling persimmons. It pollen shed obvious on a persimmon (as it is with a pecan) or is it hard to detect?

"More than one report suggests that persimmon flowers may be stimulated to set seedless fruit by pollen of other species, particularly chestnuts (but this has not been proven). It's possible yours is setting seedless fruit this way. The fact that there is some seed shows that pollen from a male persimmon is reaching it someplace. But since many persimmons, particularly ones related to Early Golden, are able to produce limbs of male flowers, you may have a male in the area withough knowing it.
Climate also has an effect. Here in Oregon "Meader" will set seedless fruit with no male around, but in many areas it must have a male to set any fruit.
It does produce limbs of male flowers, though, so a fruitless tree may suddenly start bearing after a few years, when a male limb arises. In fact, it seems that it's a survival tactic of isolated American persimmon trees to be able to produce flowers of the opposite sex in some years. Males are known that produce perfect flowers some years, and some males will produce small, seedless fruit almost every year."

On that point, I have one local native tree that produces 80% small (1") seedless fruits, and about 20% larger 1.5" fruits that usually only contain a single plump seed. I can never think to stop and look when it's blooming to figure out if it's a male or what - though I've sent scionwood to Jerry Lehman, and he says all flowers on his resulting trees were female. It's just an oddity, I guess.

I'll make a mental note to check the tree next year for any limbs which have clusters of three (male) flowers.

I haven't cut any fruit open to check for seeds. The fruit is small, but I understand Morris Burton fruit are unusually small anyway.