Binary search tree

Binary Search Tree

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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{ \kappa }[/math]
   2. left and right of type "pointer to tree item of type [math]\displaystyle{ \kappa }[/math]"
2. An object of the binary search tree type contains a pointer root of type "pointer to tree item of type [math]\displaystyle{ \kappa }[/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.

Remark

• Besides the methos of sorted sequences, binary search trees have a private method Binary Search Tree:Remove node, which receives a pointer p to a binary search tree node and removes id (possibly by removeing another node and overwriting the key to be removed with the key of the other node. Prerequisite: [math]\displaystyle{ p.left \neq void }[/math]
• 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] in these variants (because the additional operations in these methods are necessary to maintain logatihmic height are linear in the height of the tree as well=.
• For further information, see section "Binary search tree" of page Directed tree.