State
Aside from potentially improving separation of concerns, the visitor pattern has an additional advantage over simply calling a polymorphic method: a visitor object can have state. This is extremely useful in many cases where the action performed on the object depends on previous such actions.
An example of this is a pretty-printer in a programming language implementation (such as a compiler or interpreter). Such a pretty-printer object (implemented as a visitor, in this example), will visit nodes in a data structure that represents a parsed and processed program. The pretty-printer will then generate a textual representation of the program tree. To make the representation human-readable, the pretty-printer should properly indent program statements and expressions. The current indentation level can then be tracked by the visitor as its state, correctly applying encapsulation, whereas in a simple polymorphic method invocation, the indentation level would have to be exposed as a parameter and the caller would rely on the method implementation to use and propagate this parameter correctly.
Read more about this topic: Visitor Pattern
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—M. E. W. Sherwood (1826–1903)