If you have an issue with plants the first step is to ID them, then you can find the exact method to suppress them.

As to what you should do with the remains---the thing to remember is that plants are made from water and CO2. A lot of people imagine trees are made from dirt, but that's not the case. Everything other than water and CO2 is minute.

Making plants produces a lot of oxygen---decomposing them requires a lot also. But water doesn't hold much oxygen, So (TLDR) natural processes are not good at removing muck. A pitchfork however...

grower of champion bullheads