Algebraic data type

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This is a type where we specify the shape of each of the elements. Wikipedia has a thorough discussion. "Algebraic" refers to the property that an Algebraic Data Type is created by "algebraic" operations. The "algebra" here is "sums" and "products":

  • "sum" is alternation (A | B, meaning A or B but not both)
  • "product" is combination (A B, meaning A and B together)


  • data Pair = I Int | D Double is just one number, either an Int or else a Double. In this case, the tags I and D are used (in constructors and pattern matching) to distinguish between the two alternatives.
  • data Pair = P Int Double is a pair of numbers, an Int and a Double together. The tag P is used (in constructors and pattern matching) to combine the contained values into a single structure that can be assigned to a variable.

Sums and products can be repeatedly combined into an arbitrarily large structures.

Algebraic Data Type is not to be confused with *Abstract* Data Type, which (ironically) is its opposite, in some sense. The initialism "ADT" usually means *Abstract* Data Type, but GADT usually means Generalized *Algebraic* Data Type.

Tree examples

Suppose we want to represent the following tree:

             / \
            3   7
           / \
          1   4

We may actually use a variety of Haskell data declarations that will handle this. The choice of algebraic data types determines its structural/shape properties.

Binary search tree

In this example, values are stored at each node, with smaller values to the left, greater to the right.

data Stree a = Tip | Node (Stree a) a (Stree a)

and then our example tree would be:

  etree = Node (Node (Node Tip 1 Tip) 3 (Node Tip 4 Tip)) 5 (Node Tip 7 Tip)

To maintain the order, such a tree structure is usually paired with a smart constructor.

Rose tree

Alternatively, it may be represented in what appears to be a totally different structure.

data Rose a = Rose a [Rose a]

In this case, the example tree would be:

retree = Rose 5 [Rose 3 [Rose 1 [], Rose 4[]], Rose 7 []]

The differences between the two are that the (empty) binary search tree Tip is not representable as a Rose tree, and a Rose tree can have an arbitrary and internally varying branching factor (0,1,2, or more).

See also