I'm not directly aware of any trees that do that, but it honestly wouldn't surprise me if that was true for at least one species.

Most trees that drop seeds have methods of getting them away from the parent tree. Maple trees have little helicopter twirlies, oak trees have animals carry them off, some trees grow new ones from their roots called suckers.

I'll try to look it up later because now I'm curious. I'll update here if I find anything interesting about trees dropping leaves to prevent new ones growing too close.

I don't think my arborist textbook said anything about that specifically, but I'll double check there first.