A helmet should be worn at all times by every climber and everyone under the climbing tree. The obvious applies: Helmets protect the climber and people on the ground from falling objects, such as gear that is dropped or dead branches that break off. Any falling object can land in unpredictable places as it bounces off other branches on the way down.
Always use a helmet with a chinstrap to prevent it from falling off while you're looking up. If you plan to climb in the summer, get a helmet with ventilation holes.
Buy a good helmet made especially for tree climbers; bicycle or hockey helmets are not designed to protect the head from falling objects. Tree climbing helmets are available in children's and adult sizes, easy to adjust, and feature comfortable foam padding. The lower priced helmets are designed for taking a hit only on the top of the helmet. Higher quality helmets are designed for taking a hit on the top, sides, and front and back. They are the choice of more serious climbers as well as professionals. They even come in a variety of colors!
See "Gear Rater" reviews for HELMETS.