The theorized smallest amount of time is Planck time.
It's the amount of time it takes light to travel one Planck length in a vacuum.

The actual smallest amount of observable time measured as of yet was 10
26 tP's.
Therefore, a "moment" cannot be an infinitesimal period of time.
So, going back to what I was saying; The moment I was speaking of was an implied half-life event of an atomic or subatomic particle. Therefore, without recording or observing, the status of the fallen tree having made a sound or not is undeterminable and the Universe splits off at the moment, therefore both possible outcomes occurred. Only until we observe it will we know for sure which path our Universe took. As I also said, even if you placed a audio recording device nearby, it's possible that it wouldn't make a sound (the recorder gets destroyed somehow, which would mean the Universe took the path that no sound was made as nothing was able to analyze any info from the event and the data is lost forever).