Algebraic data type
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
Bbut not both)
- "product" is combination (
A B, meaning
data Pair = I Int | D Doubleis just one number, either an
Intor else a
Double. In this case, the tags
Dare used (in constructors and pattern matching) to distinguish between the two alternatives.
data Pair = P Int Doubleis a pair of numbers, an
Doubletogether. The tag
Pis 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.
Suppose we want to represent the following tree:
5 / \ 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.
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).