The Composite design pattern solves problems like:
How can a part-whole hierarchy be represented
so that clients can treat individual objects and compositions of objects uniformly?
See Applicability section for all problems Composite can solve. See Solution section for how Composite solves the problems.

• An inflexible way to represent a part-whole hierarchy is to define (1) Part objects and (2) Whole objects that act as containers for Part objects.
  Clients of the hierarchy must treat Part and Whole objects differently, which makes them more complex especially if the object structure is constructed and traversed dynamically.
• That's the kind of approach to avoid if we want to simplify client code so that all objects in the hierarchy can be treated uniformly.
• For example, representing Bill of Materials.
  A Bill of Materials (BOM) is a part-whole structure that describes the parts and subcomponents (wholes) that make up a manufactured product (see also Builder for creating a BOM).
  It should be possible, for example, to calculate the total price either of an individual part or a complete subcomponent without having to treat part and subcomponent differently (see Sample Code / Example 2 / BOM).
• For example, representing text documents.
  A text document can be organized as part-whole hierarchy consisting of characters, pictures, etc. (parts) and lines, pages, etc. (wholes).
  It should be possible, for example, to treat displaying a particular page or the entire document uniformly.