Is there a minimum spacing planted trees need to grow?

No, it's not quite like New Leaf but back to that in a sec.

Yes, you need space all around the tree/sapling, as in it can't be up against a building, right next to the slope to the beach, or even on that peninsula jutting out to the right on the island. A tree that cannot grow in that spot will just remain a stunted sapling.

A good way to test whether a tree can grow in that spot is to dig up a sapling from elsewhere and try to plant it in the target spot (rather than planting the seed or fruit). The hole won't let you plant the sapling if it's a bad location.

But back to that New Leaf stuff, the weird issue of tree spacing (in which the tree in the middle of a group of trees wouldn't grow) has been resolved:

Yes, new leaf rules still apply, you need space all around for a tree to exist. even if you uproot the whole thing using hunger, it wont let you replant the tree if there isnt room.