This will depend on your policy. It's common for policies to cover removal if a tree falls in a storm and damages the house. If the tree falls but doesn’t cause damage, the insurance policy won’t cover that. It’s your responsibility to call a tree surgeon to remove the tree. An insurance policy considers the dangers of nearby trees if they are very tall and likely to cause damage. Risky trees might result in a higher premium. An insurance company shouldn’t ask you to remove a tree, but if they do, check if the tree is protected by a preservation order.