Binary search tree: Difference between revisions

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[[Category:Checkup]]
[[Category:Videos]]
 
[[Category:Data Structures]]
[[Category:Data Structures]]
[[Category:Trees]]
[[Category:Trees]]
[[Category:Binary_Search_Tree]]
[[Category:Binary_Search_Tree]]
 
{{#ev:youtube|https://www.youtube.com/watch?v=AdhRIRgVZVw|500|right|Binary search tree|frame}}
<div class="plainlinks" style="float:right;margin:0 0 5px 5px; border:1px solid #AAAAAA; width:auto; padding:1em; margin: 0px 0px 1em 1em;">
<div style="font-size: 1.8em;font-weight:bold;text-align: center;margin:0.2em 0 1em 0">Binary Search Tree</div>
 
<div style="font-size: 1.2em; margin:.5em 0 1em 0; text-align:center">[[Sorted sequence]]</div>
 
<div style="font-size: 1.2em; margin:.5em 0 .5em 0;text-align:center">[[File:olw_logo1.png|20px]][https://openlearnware.tu-darmstadt.de/#!/resource/binary-search-tree-1938 Openlearnware]</div>
</div>
[[File:Bst.png|300px|thumb|right|Simple Binary Search tree]]
[[File:Bst.png|300px|thumb|right|Simple Binary Search tree]]
== General Information ==
== General Information ==
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# An object of the binary search tree type contains a pointer '''''root''''' of type "pointer to tree item of type <math>\mathcal{K}</math>."
# An object of the binary search tree type contains a pointer '''''root''''' of type "pointer to tree item of type <math>\mathcal{K}</math>."
# The pointer '''''root''''' points to a well-formed binary search tree. In accordance with the definition of [[Directed Tree|directed trees]], "well-formed" means that, for any node, there is exactly one [[Basic graph definitions#Paths|path]] from the root to that node.
# The pointer '''''root''''' points to a well-formed binary search tree. In accordance with the definition of [[Directed Tree|directed trees]], "well-formed" means that, for any node, there is exactly one [[Basic graph definitions#Paths|path]] from the root to that node.
# For each node <math>x</math> in the tree, no key in the left subtree of that node is greater than the key of <math>x</math>, and no key in the right subtree of <math>x</math> is less than the key of <math>x</math>.


== Remark ==
== Remark ==
* Besides the methods of [[Sorted sequence|sorted sequences]], binary search trees in the implementation chosen here have a private method [[Binary Search Tree:Remove node]], which receives a pointer <math>p</math> to a binary search tree node and removes id (possibly by removing another node and overwriting the key to be removed with the key of the other node. Prerequisite: <math> p</math>.left <math>\neq</math>void.
* Besides the methods of [[Sorted sequence|sorted sequences]], binary search trees in the implementation chosen here have a private method [[Binary Search Tree:Remove node]], which receives a pointer <math>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> p</math>.left <math>\neq</math>void.
* There are variants on binary search trees, such as [http://en.wikipedia.org/wiki/AVL_tree AVL trees] and [http://en.wikipedia.org/wiki/Red_black_tree red-black-trees], for which the height of the tree is guaranteed to be in <math>O \log{n}</math> in these variants (because the additional operations in these methods are necessary to maintain logarithmic height are linear in the height of the tree as well.
* There are variants on binary search trees, such as [http://en.wikipedia.org/wiki/AVL_tree AVL trees] and [http://en.wikipedia.org/wiki/Red_black_tree red-black-trees], for which the height of the tree is guaranteed to be in <math>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 [[Directed Tree#Binary Search Tree|binary search trees]] of page [[Directed Tree]].
* The mathematical concept behind this data structure is described in the section on [[Directed Tree#Binary Search Tree|binary search trees]] of page [[Directed Tree]].

Latest revision as of 06:14, 6 July 2015

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.