Binary search tree

In computer science, a binary search tree (BST), also called an ordered or sorted binary tree, is a rooted binary tree data structure with the key of each internal node being greater than all the keys in the respective node's left subtree and less than the ones in its right subtree. The time complexity of operations on the binary search tree is directly proportional to the height of the tree.

Binary search tree
Typetree
Invented1960
Invented byP.F. Windley, A.D. Booth, A.J.T. Colin, and T.N. Hibbard
Time complexity in big O notation
Algorithm Average Worst case
Space O(n) O(n)
Search O(log n) O(n)
Insert O(log n) O(n)
Delete O(log n) O(n)
Fig. 1: A binary search tree of size 9 and depth 3, with 8 at the root. The leaves are not drawn.

Binary search trees allow binary search for fast lookup, addition, and removal of data items. Since the nodes in a BST are laid out in such a way that each comparison skips about half of the remaining tree, the lookup performance is proportional to that of binary logarithm. BSTs are devised in the 1960s for the problem of efficient storage of labeled data and are attributed to Conway Berners-Lee and David Wheeler.

The performance of a binary search tree is dependent on the order of insertion of the nodes into the tree since arbitrary insertions may lead to degeneracy; several variations of the binary search tree can be built with guaranteed worst-case performance. The basic operations include: search, traversal, insert and delete. BSTs with guaranteed worst-case complexities perform better than an unsorted array, which would require linear search time.

The complexity analysis of BST shows that, on average, the insert, delete and search takes for nodes. In the worst case, they degrade to that of a singly linked list: . To address the boundless increase of the tree height with arbitrary insertions and deletions, self-balancing variants of BSTs are introduced to bound the worst lookup complexity to that of the binary logarithm. AVL trees were the first self-balancing binary search trees, invented in 1962 by Georgy Adelson-Velsky and Evgenii Landis.

Binary search trees can be used to implement abstract data types such as dynamic sets, lookup tables and priority queues, and used in sorting algorithms such as tree sort.

HistoryEdit

The binary search tree algorithm was discovered independently by several researchers, including P.F. Windley, Andrew Donald Booth, Andrew Colin, Thomas N. Hibbard.[1][2] The algorithm is attributed to Conway Berners-Lee and David Wheeler, who used it for storing labeled data in magnetic tapes in 1960.[3] One of the earliest and popular binary search tree algorithm is that of Hibbard.[1]

The time complexities of a binary search tree increases boundlessly with the tree height if the nodes are inserted in an arbitrary order, therefore self-balancing binary search trees were introduced to bound the height of the tree to  .[4] Various height-balanced binary search trees were introduced to confine the tree height, such as AVL trees, Treaps, and red–black trees.[5]

The AVL tree was invented by Georgy Adelson-Velsky and Evgenii Landis in 1962 for the efficient organization of information.[6][7] It was the first self-balancing binary search tree to be invented.[8]

OverviewEdit

A binary search tree is a rooted binary tree in which the nodes are arranged in total order in which the nodes with keys greater than any particular node is stored on the right sub-trees and the ones with equal to or less than are stored on the left sub-tree satisfying the binary search property.[9]: 298 [10]: 287 

Binary search trees are also efficacious in sortings and search algorithms. However, the search complexity of a BST depends upon the order in which the nodes are inserted and deleted; since in worst case, successive operations in the binary search tree may lead to degeneracy and form a singly linked list (or "unbalanced tree") like structure, thus has the same worst-case complexity as a linked list.[11][9]: 299-302 

Binary search trees are also a fundamental data structure used in construction of abstract data structures such as sets, multisets, and associative arrays.

OperationsEdit

SearchingEdit

Searching in a binary search tree for a specific key can be programmed recursively or iteratively.

Searching begins by examining the root node. If the tree is  , the key being searched for does not exist in the tree. Otherwise, if the key equals that of the root, the search is successful and the node is returned. If the key is less than that of the root, the search proceeds by examining the left subtree. Similarly, if the key is greater than that of the root, the search proceeds by examining the right subtree. This process is repeated until the key is found or the remaining subtree is  . If the searched key is not found after a   subtree is reached, then the key is not present in the tree.[10]: 290–291 

Recursive searchEdit

The following pseudocode implements the BST search procedure through recursion.[10]: 290 

 Recursive-Tree-Search(x, key)
   if x = NIL or key = x.key then
     return x
   if key < x.key then
     return Recursive-Tree-Search(x.left, key)
   else
     return Recursive-Tree-Search(x.right, key)
   end if

The recursive procedure continues until a   or the   being searched for are encountered.

Iterative searchEdit

The recursive version of the search can be "unrolled" into a while loop. On most machines, the iterative version is found to be more efficient.[10]: 291 

 Iterative-Tree-Search(x, key)
   while x ≠ NIL and key ≠ x.key then
     if key < x.key then
       x := x.left
     else
       x := x.right
     end if
   repeat
   return x

Since the search may proceed till some leaf node, the running time complexity of BST search is   where   is the height of the tree. However, the worst case for BST search is   where   is the total number of nodes in the BST, because an unbalanced BST may degenerate to a linked list. However, if the BST is height-balanced the height is  .[10]: 290 

Successor and predecessorEdit

For certain operations, given a node  , finding the successor or predecessor of   is crucial. Assuming all the keys of the BST are distinct, the successor of a node   in BST is the node with the smallest key greater than  's key. On the other hand, the predecessor of a node   in BST is the node with the largest key smaller than  's key. Following is pseudocode for finding the successor and predecessor of a node   in BST.[12][13][10]: 292–293 

 BST-Successor(x)
   if x.right ≠ NIL then
     return BST-Minimum(x.right)
   end if
   y := x.parent
   while y ≠ NIL and x = y.right then
     x := y
     y := y.parent
   repeat
   return y
 BST-Predecessor(x)
   if x.left ≠ NIL then
     return BST-Maximum(x.left)
   end if
   y := x.parent
   while y ≠ NIL and x = y.left then
     x := y
     y := y.parent
   repeat
   return y

Operations such as finding a node in a BST whose key is the maximum or minimum are critical in certain operations, such as determining the successor and predecessor of nodes. Following is the pseudocode for the operations.[10]: 291–292 

 BST-Maximum(x)
   while x.right ≠ NIL then
     x := x.right
   repeat
   return x
 BST-Minimum(x)
   while x.left ≠ NIL then
     x := x.left
   repeat
   return x

InsertionEdit

Operations such as insertion and deletion cause the BST representation to change dynamically. The data structure must be modified in such a way that the properties of BST continue to hold. New nodes are inserted as leaf nodes in the BST.[10]: 294–295  Following is an iterative implementation of the insertion operation.[10]: 294 

1    BST-Insert(T, z)
2      y := NIL
3      x := T.root
4      while x ≠ NIL do
5        y := x
6        if z.key < x.key then
7          x := x.left
8        else
9          x := x.right
10       end if
11     repeat
12     z.parent := y
13     if y = NIL then
14       T.root := z
15     else if z.key < y.key then
16       y.left := z
17     else
18       y.right := z
19     end if

The procedure maintains a "trailing pointer"   as a parent of  . After initialization on line 2, the while loop along lines 4-11 causes the pointers to be updated. If   is  , the BST is empty, thus   is inserted as the root node of the binary search tree  , if it is not  , insertion proceeds by comparing the keys to that of   on the lines 15-19 and the node is inserted accordingly.[10]: 295 

DeletionEdit

 
Fig. 2: Binary search tree special cases deletion illustration.

Deletion of a node, say  , from a binary search tree   should abide three cases:[10]: 295 

  1. If   is a leaf node, the parent node's pointer to   gets replaced with   and consequently   gets removed from the tree.
  2. If   has a single child node, the child gets elevated as either left or right child of  's parent depending on the position of   within the BST, as shown in fig. 2 part (a) and part (b), and as a result,   gets removed from the tree.
  3. If   has both a left and right child, the successor of   (let it be  ) takes the position of   in the tree. This depends on the position of   within the BST:[10]: 296 
    1. If   is  's immediate right child,   gets elevated and  's left child made point to  's initial left sub-tree, as shown in fig. 2 part (c).
    2. If   is not the immediate right child of  , deletion proceeds by replacing the position of   by its right child, and   takes the position of   in the BST, as shown in fig. 2 part (d).

Following is a pseudocode for the deletion operation in a binary search tree.[10]: 296-298 

1    BST-Delete(T, z)
2      if z.left = NIL then
3        Shift-Nodes(T, z, z.right)
4      else if z.right = NIL then
5        Shift-Nodes(T, z, z.left)
6      else
7        y := Tree-Successor(z)
8        if y.parent ≠ z then
9          Shift-Nodes(T, y, y.right)
10         y.right := z.right
11         y.right.parent := y
12       end if
13       Shift-Nodes(T, z, y)
14       y.left := z.left
15       y.left.parent := y
16     end if
1    Shift-Nodes(T, u, v)
2      if u.parent = NIL then
3        T.root := v
4      else if u = u.parent.left then
5        u.parent.left := v
5      else
6        u.parent.right := v
7      end if
8      if v ≠ NIL then
9        v.parent := u.parent
10     end if

The   procedure deals with the 3 special cases mentioned above. Lines 2-3 deal with case 1; lines 4-5 deal with case 2 and lines 6-16 for case 3. The helper function   is used within the deletion algorithm for the purpose of replacing the node   with   in the binary search tree  .[10]: 298  This procedure handles the deletion (and substitution) of   from the BST.

TraversalEdit

A BST can be traversed through three basic algorithms: inorder, preorder, and postorder tree walks.[10]: 287 

  • Inorder tree walk: Nodes from the left subtree get visited first, followed by the root node and right subtree.
  • Preorder tree walk: The root node gets visited first, followed by left and right subtrees.
  • Postorder tree walk: Nodes from the left subtree get visited first, followed by the right subtree, and finally the root.

Following is a recursive implementation of the tree walks.[10]: 287–289 

 Inorder-Tree-Walk(x)
   if x ≠ NIL then
     Inorder-Tree-Walk(x.left)
     visit node
     Inorder-Tree-Walk(x.right)
   end if
 Preorder-Tree-Walk(x)
   if x ≠ NIL then
     visit node
     Preorder-Tree-Walk(x.left)
     Preorder-Tree-Walk(x.right)
   end if
 Postorder-Tree-Walk(x)
   if x ≠ NIL then
     Postorder-Tree-Walk(x.left)
     Postorder-Tree-Walk(x.right)
     visit node
   end if

Balanced binary search treesEdit

Without rebalancing, insertions or deletions in a binary search tree may lead to degeneration, resulting in a height   of the tree (where   is number of items in a tree), so that the lookup performance is deteriorated to that of a linear search.[14] Keeping the search tree balanced and height bounded by   is a key to the usefulness of the binary search tree. This can be achieved by "self-balancing" mechanisms during the updation operations to the tree designed to maintain the tree height to the binary logarithmic complexity.[4][15]: 50 

Height-balanced treesEdit

A tree is height-balanced if the heights of the left sub-tree and right sub-tree are guaranteed to be related by a constant factor. This property was introduced by the AVL tree and continued by the Red-Black tree.[15]: 50–51  The heights of all the nodes on the path from the root to the modified leaf node have to be observed and possibly corrected on every insert and delete operation to the tree.[15]: 52 

Weight-balanced treesEdit

In a weight-balanced tree, the criterion of a balanced tree is the number of leaves of the subtrees. The weights of the left and right subtrees differ at most by  .[16][15]: 61  However, the difference is bound by a ratio   of the weights, since a strong balance condition of   cannot be maintained with   rebalancing work during insert and delete operations. The  -weight-balanced trees gives an entire family of balance conditions, where each left and right subtrees have each at least a fraction of   of the total weight of the subtree.[15]: 62 

TypesEdit

There are several self-balanced binary search trees, including T-tree,[17] treap,[18] red-black tree,[19] B-tree,[20] 2–3 tree,[21] and Splay tree.[22]

Examples of applicationsEdit

SortEdit

Binary search trees are used in sorting algorithms such as tree sort, where all the elements are inserted at once and the tree is traversed at an in-order fashion.[23] BSTs are also used in quicksort.[24]

Priority queue operationsEdit

Binary search trees are used in implementing priority queues, using the node's key as priorities. Adding new elements to the queue follows the regular BST insertion operation but the removal operation depends on the type of priority queue:[25]

  • If it is an ascending order priority queue, removal of an element with the lowest priority is done through leftward traversal of the BST.
  • If it is a descending order priority queue, removal of an element with the highest priority is done through rightward traversal of the BST.

See alsoEdit

ReferencesEdit

  1. ^ a b Culberson, J.; Munro, J. I. (1 January 1989). "Explaining the Behaviour of Binary Search Trees Under Prolonged Updates: A Model and Simulations". The Computer Journal. 32 (1): 68–69. doi:10.1093/comjnl/32.1.68.
  2. ^ Culberson, J.; Munro, J. I. (28 July 1986). "Analysis of the standard deletion algorithms in exact fit domain binary search trees". Algorithmica. Springer Publishing, University of Waterloo: 297. doi:10.1007/BF01840390.
  3. ^ P. F. Windley (1 January 1960). "Trees, Forests and Rearranging". The Computer Journal. 3 (2): 84. doi:10.1093/comjnl/3.2.84.
  4. ^ a b Knuth, Donald (1998). "Section 6.2.3: Balanced Trees". The Art of Computer Programming (PDF). Vol. 3 (2 ed.). Addison-Wesley. pp. 458–481. ISBN 978-0201896855.
  5. ^ Paul E. Black, "red-black tree", in Dictionary of Algorithms and Data Structures [online], Paul E. Black, ed. 12 November 2019. (accessed May 19 2022) from: https://www.nist.gov/dads/HTML/redblack.html
  6. ^ Myers, Andrew. "CS 312 Lecture: AVL Trees". Cornell University, Department of Computer Science. Archived from the original on 27 April 2021. Retrieved 19 May 2022.
  7. ^ Adelson-Velsky, Georgy; Landis, Evgenii (1962). "An algorithm for the organization of information". Proceedings of the USSR Academy of Sciences (in Russian). 146: 263–266. English translation by Myron J. Ricci in Soviet Mathematics - Doklady, 3:1259–1263, 1962.
  8. ^ Pitassi, Toniann (2015). "CSC263: Balanced BSTs, AVL tree" (PDF). University of Toronto, Department of Computer Science. p. 6. Archived (PDF) from the original on 14 February 2019. Retrieved 19 May 2022.
  9. ^ a b Thareja, Reema (13 October 2018). "Hashing and Collision". Data Structures Using C (2 ed.). Oxford University Press. ISBN 9780198099307.
  10. ^ a b c d e f g h i j k l m n o p Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). Introduction to Algorithms (2nd ed.). MIT Press. ISBN 0-262-03293-7.
  11. ^ R. A. Frost; M. M. Peterson (1 February 1982). "A Short Note on Binary Search Trees". The Computer Journal. Oxford University Press. 25 (1): 158. doi:10.1093/comjnl/25.1.158.
  12. ^ Junzhou Huang. "Design and Analysis of Algorithms" (PDF). University of Texas at Arlington. p. 12. Archived (PDF) from the original on 13 April 2021. Retrieved 17 May 2021.
  13. ^ Ray Toal. "Binary Search Tree". Loyola Marymount University, Department of Computer Science. Retrieved 17 May 2022.
  14. ^ Thornton, Alex (2021). "ICS 46: Binary Search Trees". University of California, Irvine. Archived from the original on 4 July 2021. Retrieved 21 October 2021.
  15. ^ a b c d e Brass, Peter (January 2011). Advanced Data Structure. Cambridge University Press. doi:10.1017/CBO9780511800191. ISBN 9780511800191.
  16. ^ Blum, Norbert; Mehlhorn, Kurt (1978). "On the Average Number of Rebalancing Operations in Weight-Balanced Trees" (PDF). Theoretical Computer Science. 11 (3): 303–320. doi:10.1016/0304-3975(80)90018-3.
  17. ^ Lehman, Tobin J.; Carey, Michael J. (25–28 August 1986). A Study of Index Structures for Main Memory Database Management Systems. Twelfth International Conference on Very Large Databases (VLDB 1986). Kyoto. ISBN 0-934613-18-4.
  18. ^ Aragon, Cecilia R.; Seidel, Raimund (1989), "Randomized Search Trees" (PDF), Proc. 30th Symp. Foundations of Computer Science (FOCS 1989), Washington, D.C.: IEEE Computer Society Press, pp. 540–545, doi:10.1109/SFCS.1989.63531, ISBN 0-8186-1982-1
  19. ^ Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald L.; Stein, Clifford (2001). "Red–Black Trees". Introduction to Algorithms (second ed.). MIT Press. pp. 273–301. ISBN 978-0-262-03293-3.
  20. ^ Comer, Douglas (June 1979), "The Ubiquitous B-Tree", Computing Surveys, 11 (2): 123–137, doi:10.1145/356770.356776, ISSN 0360-0300, S2CID 101673
  21. ^ Knuth, Donald M (1998). "6.2.4". The Art of Computer Programming. Vol. 3 (2 ed.). Addison Wesley. ISBN 9780201896855. The 2–3 trees defined at the close of Section 6.2.3 are equivalent to B-Trees of order 3.
  22. ^ Sleator, Daniel D.; Tarjan, Robert E. (1985). "Self-Adjusting Binary Search Trees" (PDF). Journal of the ACM. 32 (3): 652–686. doi:10.1145/3828.3835. S2CID 1165848.
  23. ^ Narayanan, Arvind (2019). "COS226: Binary search trees". Princeton University School of Engineering and Applied Science. Archived from the original on 22 March 2021. Retrieved 21 October 2021 – via cs.princeton.edu.
  24. ^ Xiong, Li. "A Connection Between Binary Search Trees and Quicksort". Oxford College of Emory University, The Department of Mathematics and Computer Science. Archived from the original on 26 February 2021. Retrieved 4 June 2022.
  25. ^ Myers, Andrew. "CS 2112 Lecture and Recitation Notes: Priority Queues and Heaps". Cornell University, Department of Computer Science. Archived from the original on 21 October 2021. Retrieved 21 October 2021.

Further readingEdit

External linksEdit