### Introduction to Recursive Types

 Many functional languages, especially those in the ML family, include algebraic data types (discriminated union types), which can be recursive. These types provide an effective, light-weight mechanism for creating tree-like structures of arbitrary depth. Behavior can be added easily in the form of recursive functions that use pattern matching for branching on the structure of the argument. These types are closed in the sense that it is not possible to add new variants to the type.By contrast, in mainstream object-oriented languages, one would typically create an interface to represent the type and a class for each variant of the type, using the Composite and Decorator patterns for structure and, sometimes, the Visitor pattern for behavior. Without the visitor, it is very easy to add more variants; with the visitor, this is still possible but harder.This discussion provides more background on this trade-off. Organizational charts (org charts) are a simple, well understood example of a hierarchical structure. The following example explores how we can represent and work with org charts in F#. In this example, an org chart is either a person (a leaf in the resulting tree) or an organizational unit (an interior node), which has zero or more org charts as children.> type OrgChart = Person of string | OrgUnit of string * OrgChart list;; type OrgChart =   | Person of string   | OrgUnit of string * OrgChart list > Person "George";; val it : OrgChart = Person "George" > let cs = OrgUnit ("cs", [ Person "Sekharan"; Person "Rom"; Person "Thiruvathukal" ]);; val cs : OrgChart =   OrgUnit ("cs",[Person "Sekharan"; Person "Rom"; Person "Thiruvathukal"]) > let math = OrgUnit ("math", [ Person "Jensen"; Person "Doty" ]);; val math : OrgChart = OrgUnit ("math",[Person "Jensen"; Person "Doty"]) > let cas = OrgUnit ("cas", [ Person "Fennel"; cs ; math ] );; val cas : OrgChart =   OrgUnit     ("cas",      [Person "Fennel";       OrgUnit ("cs",[Person "Sekharan"; Person "Rom"; Person "Thiruvathukal"]);       OrgUnit ("math",[Person "Jensen"; Person "Doty"])]) > let luc = OrgUnit ( "luc", [ cas ] );; ...Now we will define a size function on org charts. If the org chart is a person, then the size is one. Else we compute the size recursively from the list of children, orgs. (Each item in this list can be a person or OU, but this distinction will be made in the recursive application of the function to each child.) Now we apply the size function recursively to each child of orgs to find out the size of each child. Instead of explicit loops, we use List.map to apply a function to each item in a list and List.fold for adding up the results. (These functions are discussed in a separate handout in more detail.)> let rec size = function  | Person _ -> 1          | OrgUnit ( _, orgs ) -> List.fold (+) 0 (List.map size orgs);;val size : OrgChart -> int > size math;; val it : int = 2 > size cs;; val it : int = 3 > size luc;; val it : int = 6