What is a full breath? So go a head and take the biggest breath in you can and let it out. Do you breathe in through your mouth? Did you breath direct into your chest? Or did you breathe in through your nose slowly filling your diaphragm(belly) then laterally into your ribs then up into your chest? If so you are among the few as nearly 50-60% of most people breathe in through their mouth and directly into their chest not accessing the full potential of respiratory system; diaphragm, as well as the muscles of the ribs, back, and chest. Your nose is the regulator for your entire system. Do you have hair growing in the back of your throat? Of course not. That's why you have hair in your nose to filter the hair and warm it. Your mouth is for eating your nose is for breathing. Modern man has lost his connection to the breath.
The video above is a basic breathing exercise to train the entire breathing system to work together. It's broken down into 3 sections diaphragm, ribs, and chest. Most people have difficulty breathing slowly into their nose filling the diaphragm. Which is the primary pump or plunger that pulls air into the lungs. I said plunger like a toilet plunger, because the diaphragm is like an upside down toilet plunger that has been flattened so when the diaphragm contracts(unflatten) it expands down pulling the air into the lungs like the toilet plunger pulls out the clog when it un-flattens. Don't worry if its difficult but don't stop training it. As you become better at this the better you will breath.
Now the expansion of the ribs is not something many people are aware of. There is so much room in there to fill your lung to capacity. Just be conscious to breathe your ribs laterally across the floor and not vertically to the ceiling putting your back into extension or arching over the floor. Their are so many muscle in the ribs that aid with breathing but then again modern man's posture has collapsed and these muscle have gotten shorter and tighter from this collapse. So let train them so you breathe better and move better.
Lastly the chest is well known but there are the muscles of the back between the shoulder blades that can expand and pull more air into the upper lungs. Learning to breathe and expand these muscles will not only improve your breathing but also your posture.
Actually these exercise will definitely improve your posture a bit as all three zones (diaphragm, ribs and chest) have muscle responsible for holding you upright in good posture, with good posture comes good movement. I'm willing to say good breathing and posture can improve performance.