At least not commonly (I have a vague recollection that I may have seen something like this but it’s certainly uncommon at least).
My next door neighbor has a big locust type tree which overhangs my yard, and it drops down seedlings in my yard. I either rip these up or (more commonly) just mow them over, but they keep growing back. I can tell it’s the same ones, because the roots are pretty strong and not waspy-frail like new seedlings.
But there was one that I decided to keep for a while. I eventually cut it down when it got about 15-20 feet tall, and at that point it was done. I left a stump about 6 inches tall, which is a lot taller than the smaller ones that I mow over, but the stump didn’t regrow, and in fact disintegrated over the next year or two (to the point where I poked a shovel under it and popped it off the roots). And this has been my experience with larger stumps that I’ve had or seen - they don’t regrow and just disintegrate instead.
It varies from species to species. Some tree types are very good at putting up shoots from stumps. That’s the whole basis of coppicing. But not all trees coppice well.
Many species of trees do stump-sprout, and can regenerate a new tree after being cut down. It depends on the ecology of the species, and also may depend on individual conditions. Some species of trees, like quaking aspens, regularly reproduce by putting up new trunks from their extended root system. In the Rockies, entire patches of aspen forest can be essentially a single clonal individual that sprouted from the same root system.
There is a single Aspen tree (ok Colony but it’s the same single organism) in Fishlake National forest believed to be the world’s largest organism at 6 million kilograms and about 80,000 years old.
For a single variety of tree like in the OP, the main cause might simple be old vs. new wood. On an older tree, the bark around the stump (and the main roots near it) is old and tough. The sites that green wood has that can produce new growth are less common and have a tougher time regenerating.
I was anticipating a lot of sprouts with the big trees I had cut down a few years ago. Nada.
I don’t know if it outweighs the Aspen colony, of if they are using different criteria for what constitues a single organism, but on some measure I’ve seen the largest organism reported as the honey fungus Armillaria solidipes.
The article isn’t clear on the criteria. It’s surely not sufficient for above-ground occurrences to be genetically identical, clones are not the same organism. Presumably they have evidence that the fungus is a single linked organism underground.
ETA: there’s a Wikipedia article that discusses it relative to the Aspen colony:
I remember when I was farming we used to use chestnut branches as posts for fencing. Chestnut trees were abundant in my area and the wood, being very resistant to rot, was much used in this regard. Occasionally, to our amazement, one of these fenceposts would sprout.