Binary search tree

Binary search tree

Simple Binary Search tree

General Information

Abstract data structure: Sorted sequence

Implementation invariant:

1. There is a tree item type with three components:
   1. key is of generic type [math]\displaystyle{ \mathcal{K} }[/math],
   2. left and right of type "pointer to tree item of type [math]\displaystyle{ \mathcal{K} }[/math]."
2. An object of the binary search tree type contains a pointer root of type "pointer to tree item of type [math]\displaystyle{ \mathcal{K} }[/math]."
3. The pointer root points to a well-formed binary search tree. In accordance with the definition of directed trees, "well-formed" means that, for any node, there is exactly one path from the root to that node.
4. For each node [math]\displaystyle{ x }[/math] in the tree, no key in the left subtree of that node is greater than the key of [math]\displaystyle{ x }[/math], and no key in the right subtree of [math]\displaystyle{ x }[/math] is less than the key of [math]\displaystyle{ x }[/math].

Remark

• Besides the methods of sorted sequences, binary search trees in the implementation chosen here have a private method Binary Search Tree:Remove node, which receives a pointer [math]\displaystyle{ p }[/math] to a binary search tree node and removes it (possibly by removing another node and overwriting the key to be removed with the key of the other node. Prerequisite: [math]\displaystyle{ p }[/math].left [math]\displaystyle{ \neq }[/math]void.
• There are variants on binary search trees, such as AVL trees and red-black-trees, for which the height of the tree is guaranteed to be in [math]\displaystyle{ O(\log{n}) }[/math] at any time (because the additional operations to maintain logarithmic height are linear in the height of the tree as well).
• The mathematical concept behind this data structure is described in the section on binary search trees of page Directed Tree.