After putting a cloth on the branch, you'll need a cutting tool to cut it. So put a villager on the cutting tool in the research lab. Then, put a villager on the soap bars to clean the cut.

If your villager has died and you needed him to complete the puzzle that would mean that you'd never finish it, which seems very unfair. If you are not sure what to use to cut the branch, what do the villagers generally use for cutting? Hint: most people will have solved this as their first puzzle.

Tree limb regrow as shown in the puzzle's picture? That would be neat...

Yes - but not like any tree you ever saw.. New sprout should be be growing from just inside bark, not from core. (I had to remove screenshot as I couldn't figure out how to get it inside 'spoiler'.)

Because the inner wood is actually dead in real trees. lol. How long does it take?

it was after solving puzzle #16

That usually requires a villager to meditate. To make a villager meditate, drag him to the area between the wind flutes. I didn't have to make a villager meditate to complete this puzzle.