Both timed breathing and overbreathing are covered in "The Encyclopedia of the Pivot System". Must reading for any serious trumpet player. Now for the full breath on every note or phrase. Have you ever had to play 1 note by itself to fill out a chord in a song? What a the 3 or 4 measure phrases? These do not require as much air as a full 8 measure phrase. At the end of a very short phrase an inexperienced brass player will feel a need to exhale before he or she can take a breath. If this overbreathing continues for any length of time the player will sometimes turn red or gasp for air. No you didn't run out of air for playing however, your body NEEDS to have oxygen in your lungs. What has happened is you took a full breath and used less than half. Now when you take a full breath you only replace half of the stale oxygen depleted air in your lungs. As this continues you end up gasping for air. Does this sound familiar?
Overbreathing really is a kind of self suffication (in the extreme).
In the extreme upper register overbreathing becomes more apparent. Have you seen people get dizzy, lightheaded, or blackout. They were overbreathing. I know some people say if you release the pressure really slowly it will not happen. If you did not overbreathe and have so much leftover air under pressure it would not happen either.
Timed breathing is another aspect of playing. Some people always take a deep full breath. When playing in the upper register this creates tension. The upper register takes air compression and speed but not air mass. The low notes need the full breaths. Try a half or quarter breath before you play your next high g. This will allow your muscles to do their job.