There exist two camps in the realm of breathing.....
1. Take as much as you need
2. Take a full breath
People, especially on the internet, are often confused by each of these; which method is better than the other, what is the preferred method, how do you know what to do, etc. Yet, all it takes is a simple example: Take as much air in to speak and play a middle C, short. Then, take a full breath and play the same, short pitch. Which sounds better? The one with the full breath! Why? Because you don't need to "muscle" the note out. Your air already contains the power necessary and you achieve a better, deeper, resonating sound. Can you play by "taking as much as you need"? Of course you can, but to achieve the best sounding note/s possible, a full breath AND EXHALING INTO THE INSTRUMENT is always the way to go.
What about "over breathing"? Well, simply put, don't do it - yes, I say this "tongue and cheek". If you take a full breath, then use it. Don't blow with a thin, unsupported air column - or don't engage your muscles (embouchure, bodily muscles) so much you end up choking off the air. Engagement is necessary, but we are not lifting weights here, too much muscle is a bad thing. You need to have relaxation within the engagement. Allow the air to flow/blow through the instrument (don't blow AT the instrument). What it comes down to is breath control, controlling the airflow in a way where you stay engaged but do not exaggerate the engagement. How do you know if it's correct? Listen to your sound. It should be free of over tension, round, resonant, and beautiful. Will this feeling change in other registers? Absolutely, but the sound is the key. The sound quality should never change.
So, how do we breathe? Ask 100 people and you may get 100 answers, its unbelievable. But, as I always try to do with my students, let's keep things easy... breath-in mimicking the OH sound, and exhale mimicking the HO sound. Take in as much air, OH, as your body will allow, without pain, without constricting the muscles/tension. Now, turn the air around and blow, HO, using warm air (some people may feel this coming from the throat). Keep the air stream consistent and moving. Use some simple flow studies by Chicowitz or Stamp, scales, or simple melodies (all slurred for connection purposes).
And what about higher notes? Well, for me, its more of a mental game than a physical one. If you play like you read music, the high notes are high and low notes are low (hence, their name). But if you lay the music flat, you immediately notice that lower notes are closer to you and higher notes are farther away. Which brings us full circle to breath control.... my lower pitches do not need fast air and my higher pitches do not need slow air. I will need to control my air speed for each register and the control needs to be balanced, not forced. I blow closer to the bell for lower notes and farther from the bell for higher notes. Just thinking in this manner has freed my playing up, immensely.
I will always have people tell me that they don' need air for this or for that.... ok, fine. Then lets name our instrument something other than a wind instrument.
Its simple, really - our air does SO much for our playing. Let's keep it simple and let's keep it real...