As someone once yelled to me when they saw me running down from the Longs Peak Boulder Field: "Hey, that's dangerous!". Indeed.
I've had some thoughts on this over the last couple of years, and here they are:
1) Adapt to the trail conditions. Trail running is three-dimensional and is subject to weather and time of day, so your running has to adapt to these conditions. Sometimes you run fast, sometimes you trot slowly down rocks, sometimes you walk. Allow your gait to submit to what the trail hands to you. If you don't observe this rule, the trail can reach out and slap you.
2) Get your toes out of the way. Running with toes swinging forward near to the ground is going to get you stubbed toes and scraped knees. I lift my heel up with my toes trailing and foot relaxed, then swing my leg forward with the knee still high, to be sure my foot clears rocks and roots.
3) Observe the 4-inch Landing Rule. I have a rule that I won't let my foot land less than about 4 inches behind an obstacle. This gives me a margin of safety in case I land a bit off or I'm getting tired and not picking my feet up, so I don't trip.
4) Avoid running when your balance is off. For me, this has usually been after a bad night's sleep, but it could be due to a head cold, medication, etc. Just don't. Walk that day, enjoy the view and take pictures.
5) Slow down at altitude. To some extent this takes care of itself because you simply can't run as fast with less oxygen, but if you are at a high elevation, especially if you are fatigued and/or you find yourself starting to stumble or catch your toe, slow down. If you are way out on trail, it's going to be a tough trip back with a sprained ankle.
6) Relax. It amazes me how much this applies to all athletic activity, but it's especially useful for running in the mountains. Use only the energy you need to use; you will cramp less, go farther, and feel better. Learning to run fast while doing this is a great way to learn efficiency, because you cannot move fast for long unless you are efficient.