In short, no, you should not spray your tree. There are active treatments that use what is called bark banding, spraying the trunk 6ft down. This protocol is relatively safer than full tree sprays and much less invasive to the environment.
Most full-canopy tree sprays are non-selective insecticides (meaning they kill all insects to include beneficials and parasites). This doesn’t mean that there isn’t a time and place for it. Major consideration should be made into why there are insect problems to begin with. Rather than non-selective sprays, we often recommend preventative insect and mite treatments that are more targeted and effective.
We prefer not to spray trees; most of the time, the benefits don’t outweigh the bad that comes with it. We always weigh the pros and cons of organic tree treatments vs chemical usage to find the healthiest path forward. For many homeowners, a long-term organic tree care program is the best way to keep trees strong enough to fight off pests naturally.