Adapted from my answer to Correct breathing while bouldering with slight modifications for dealing with differences between bouldering (more stop-and-go, brief high intensity moves) and high inclination hiking (more steadily challenging, with longer stops when one does take a break).
In various sports and even day-to-day life, breathing techniques make a difference in performance. Generally, you want to breathe in your nose and out your mouth, and breathe deeply, steadily, and with your breathing muscles relaxed. Deep, steady, relaxed breathing can be especially difficult during exercise. You may need to adapt your rate of breathing depending on your activity level. Nonetheless if you can breathe deeply and steadily (rhythmically) that will help maintain performance and stay relaxed.
It is important not to hold your breath because you put a lot of strain on yourself that way, and you can injure yourself more easily. Regarding particularly strenuous movements when you really have to haul your weight around, Ben C. noted the torso tension that can naturally be produced in weight lifting by holding one's breath, but weight lifters will also train to breathe through lifts for the sake of their long-term health and performance. One way to achieve the positives of both regularly breathing + holding your breath is to time your motions with your exhale or inhale. This is common in martial arts and yogas.
You can familiarize yourself with controlled breathing by practicing it with push-ups or other simple movements. Here's an example routine:
- While breathing calmly and with a resting heart rate, enter push-up position
- Begin each push-up movement with an inhale or exhale. Go down with an inhale, go up with an exhale (or vice versa, up to you, or perhaps mix it up and move on inhale before noon and move on exhale after noon). Each breath begins slightly before the motion, as if the motion follows the breath / the breath leads the motion.
- Continue exercising this way until your breathing and heartbeat is in an excited state. At that point, relax and try to return to a resting state as soon as possible. Lay down, sit, stand, pace - whatever you need, but especially breathe with your entire lungs (from chest to belly) - to return to a calm, comfortable breath and heart rate.
- Once you're back at a resting state, begin the practice again. As for how many times to go through this cycle, that's up to you, but I recommend you do fewer 'reps' in each session but have more frequent sessions (eg. only go through the cycle 3 times in a session, but have practice sessions many times a week). You can also mix up the exercise with push-ups one day, squats another day, a climbing activity another instance, taichi another instance, etc. As noted, this kind of practice can be integrated into everday-day activities - something as simple as opening a door or picking up & putting down a bag can be an opportunity to practice synchronizing breath and movement.
This routine of controlled excitation then relaxation is used by professional athletes (and was mentioned in a TGO.SE Q&A about scuba) as a way to develop more control over your breath, greater lung capacity, and importantly, more relaxed breathing during times of stress and exercise. As this aids in the performance of martial artists, yoga practitioners, and olympic athletes, I think it will also aid you in climbing.
A specific example of how this would be applied in tougher hikes:
As you begin and work through the hike, maintain a rhythmic, deep breath that uses your entire lungs, from chest to belly. When you move in a way that takes extra 'umph', make sure you synchronize the movement with a strong exhale or inhale. You should do this for all movements if possible (e.g. synchronizing a set of X amount of steps with each breath) but especially for moves that take extra energy or courage. You would begin a movement with eg. inhaling - a moment after your inhale begins you would begin the movement - with the inhale and the intended movement completing around the same time.
Throughout the hike, try to keep steady breathing that uses your entire lungs (and accordingly, you'll need steady or smooth movements to match that). You'll find you need to keep your body more relaxed and your motions more rhythmic. The 'relaxed' part is probably the most difficult, because you may tend to tense up muscles in use and put more strain than needed on the main muscle you're using. A key to relaxing is putting minimal tension on muscles - only enough to do what needs to be done - and also working with your entire body to make efficient, balanced movements to 'distribute' muscle tension and keep you most relaxed overall. To do this, try to pay attention to your muscle use. Ask yourself, where am I tense? What muscles am I using, and which ones do I need to use? If you find a part of your body tense that isn't actively being used, see if you can let go of the tension with an exhale, or try to move it around and loosen the muscle up a little (this can even be done while still in motion).
Posture is also very important for relaxed breathing and its benefits during hikes, and posture comes with numerous other benefits of efficient biomechanics. Even during breaks, posture is important so your lungs can take full and deep breaths, and so that you aren't unnecessarily straining muscles. Spread out! Open up! Don't get crunched. There are various websites on ideal posture for hiking which you can Google Search, but the important takeaway is pay attention to your body and let your own muscles recommend the fine-tuning. Again, practice it more often than not and practice it in everyday life to make it work in a time of need: Where am I putting my weight? Am I putting energy or tension into muscles I don't need to? Can I use my skeleton to provide support I'm currently getting from tense muscles? This fine-tuning will save you lots of energy and stress over the long haul.