Often the component-parent link is used for a Chain of Responsibility.
Decorator is often used with Composite. When decorators and composites are used together, they will usually have a common parent class. So decorators will have to support the Component interface with operations like Add, Remove, and GetChild.
Flyweight lets you share components, but they can no longer refer to their parents.
Iterator can be used to traverse composites.
Visitor localizes operations and behavior that would otherwise be distributed across Composite and Leaf classes.