When you add the carbon matter to your compost you want it to be shredded- you want it to be relatively small in size. The reason why is that the more surface area it has, the more surface that the microbes can feed on. So if you put some large branches in here, it would take a long time for those to break down because the microbes can only get to the outer layer. But if you shred those branches, it will break down in a couple of months.
The way you would want to shred or chop organic matter from the trimmings around your yard is with a shredder. You can also run over it with a lawn mower- like old leaves, that works great, too. But as far as the branches and things, there’s not really a good way, practically speaking, to get those broken down unless you have a shredder.