I've allowed my coconut resource to "dry up". Any way I can get the villagers to encourage new growth on the trees, or are the coconuts gone for good?
Thanks!!

Once the coconuts are gone, they will never replenish in sufficient quantities to support your villagers. You will need another source of food. If you need help, there are hints and spoilers in the Guide Section at the bottom of the forum list.

But I've noticed they replenish a little bit without events!

Yes, they grow a few extra coconuts periodically, but not enough to feed the villagers. Picking up mushrooms provides more food than that.

Coconuts replenish when crops replenish. You'll get about 30 coconuts every time that happens.

So, if you don't have any other food sources, you need to get busy on that!

IIRC, there is an event that will replenish the trees more than the normal "harvest time." But I agree with others who said that other sources of food need to be found.

Yes, it's a wind that replenishes the crops and coconuts. The times I've had this particular event clicking on the trees shows 1700 nuts awaiting harvest (the usual 1500 plus 200 extra).

I have had that particular wind, and it does fully replenish your coconuts, and for me gives the +200 as well, so it's an event that you need to happen here to fully replenish your coconuts.