The tree drops branches All big trees shed branches. This is especially true with dead branches. This job should be performed about every three years or when needed. Sure it costs money but remember that trees are valuable possessions like your home. Keeping your trees means a better resale price when your home is sold.
Some trees ARE unpredictable when it comes to dropping green branches.
If your tree has a history of dropping branches have an arborist take a look at the tree. There ARE dangerous trees out there that need to be removed. But don't just call a tree cutter because he might just need a job and persuade you to perform a needless tree removal. Call a certified arborist instead. A tree removal contractor advised the tree should come down.