The ornaments do not change the tree itself which is recognizable as a Christmas tree regardless of particular ornaments used. Adapter provides a different interface to its subject. Proxy provides the same interface. Decorator provides an enhanced interface. Adapter changes an object's interface, Decorator enhances an object's responsibilities.
Decorator is thus more transparent to the client.
As a consequence, Decorator supports recursive composition, which isn't possible with pure Adapters. Composite and Decorator have similar structure diagrams, reflecting the fact that both rely on recursive composition to organize an open-ended number of objects. A Decorator can be viewed as a degenerate Composite with only one component. However, a Decorator adds additional responsibilities - it isn't intended for object aggregation. Decorator is designed to let you add responsibilities to objects without subclassing.