Also, how many fruit cycles will the trees have?

And how often does the honey refill?

and the title question

Can a tree be chopped down?

Well, the trees will produce fruit forvever.

I don't think a tree can be chopped down.

No, trees cannot be chopped down. And yes, the fruit trees replenish forever.

I was just curious...why do you want to chop the tree? Maybe you can add the idea to the VV4 suggestion thread.

The fruit will reappear at intervals throughout the game, but it will never get to the amount that the tree started with. That doesn't include the effects of possible Island events, of course. I don't know how often the fruit or honey replenish but I suspect it partly depends on the game difficulty and speed settings. I do know they will replenish more often if you take the Nature path. Trees can't be chopped down.

I wanted to get one of each kind

Too late now, but if the fruit that drops isn't the right one, just leave it on the ground.

My tree and honey are not reproducing. Not sure what i am doing/done wrong.

Or bring a villager to take it then drop him again and he'll let the fruit falling!

They will replenish, but only infrequently. I doubt you're doing anything wrong.