We prune trees for ourselves, not for the health of the tree.
The key to tree health is to know that when we prune, a wound is created. Tree wounds don't heal, but instead the tree compartmentalizes the wound (covers with woundwood). So always remember, a wound, is a wound, is a wound!
So how do we minimize negative impacts to our tree(s)? Proper pruning and pruning at a time when all the environmental factors (heat/drought) we can't control are not present.
Step 1: Why and what am I pruning?
The illustration below provided by the A&M Texas Forest Service shows some of the reasons to prune.

Other reasons may be clearance, such as but not limited to 15 foot vertical clearance over roadways and 8 foot clearance over pedestrian areas. These heights are set by City Code.
Step 2: Best time to prune?
In Dallas, Texas the “best time” would be during our winter months as the temperatures are cooler, normally more rainfall and less insect activity. All of these factors decrease stress on the tree.
But this is not always possible….if you have a broken branch that is at risk of causing damage or harm, remove it as soon as possible. Below is a diagram how you can make the proper cut and give your tree the best chance for the woundwood to cover and compartmentalize the wound.
Step 3: How to prune
See the bulge in the picture below? That’s the branch collar!
Now what? See the location of the branch collar in the illustration below, you cut just beyond.

Illustration provided by A&M Texas Forest Service