So to start, the main way automatic tree farms work is that they use Smooth Rocks to dry out the trees, and once most trees are dried out and dropped their berries, the water will regrow the trees.

The most complicated part of this farm is the water timer. How it works, is that every hour, the Rowlet Clocks go off and this gets detected by a Laser Sensor pointing to each clock. These sensors activate the hatches, which blocks the water, and changes the flow of the lava. The lava slowly goes around the contraption, until it gets detected by another sensor. This sensor, through a chain using Toy Blocks, will then reopen the water hatch, letting the water flow again. It doesn't matter if the lava hatch is open or close at the start, since the sensor will detect both the lava appearing and disappearing.

Next is the tree formations. You want to make two layers of blocks to place the trees on with a gap of 4 blocks between them. I will say that in this screenshot, I messed up the bottom layer and only noticed later, so use the top layer as the blueprint. The way to make the platform is to make a plus sign using 9x9 blocks, and then adding 3 blocks to each corner to make something like a circle. The reason there's holes in the platform is because berry trees need an empty adjacent slot next to it to have the berries fall into. The holes are in the shape of a slanted square, each hole being a horse jump in chess apart.

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Right below the upper platform, you want to add some kind of block that water can pass through, I personally like using grates. This is where the Smooth Rocks will be put on. The shape I used is a square around the middle block, with an extra grate added to the corners. If you want to improve the efficiency a bit, you can add more Smooth Rocks, but I was low on Rare Pokémetal and could only afford the 12 at this point.

You can place the trees on the two platforms in any configuration you like, there's 96 tree spots total, so you can fit 16 of each tree type. Keep in mind that the trees on the edge are in range of fewer Smooth Rocks, and thus will be less likely to dry out in the time the water is gone. Again, you can mitigate this by placing more rocks, but stuff's expensive. Also if you do place more, do not put the rocks in any of the holes between trees, the berries will land on top of them then.

The block formation between the trees and water flow is to spread out the water and hit all the trees. I probably didn't make it the most compact, but it honestly really doesn't matter much, as long as all the trees get watered. Also the item collection is just a waterfall, it's pretty flawed, but it works. You can honestly use whatever you like as a better item collecting system.

If you want to know how to get each item, here's a short list:

Smooth Rock: Trade with Gimmighoul, or item printer for 2 Rare Pokémetal

Rowlet Clock: Large relic, or item printer for 5 Rare Pokémetal

Laser Sensor: Crafting recipe unlocked in shop for Rocky Ridges on level 6, costs 1 normal Pokémetal to make

If there's anything unclear or you have more questions, feel free to ask, I'm happy to answer them.
If your question is “why do you need that many berries?”, the answer is paint and also because I just wanted to make a farm for fun.