Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems.... , a binary search tree (BST) is a binary tree
Binary tree
In computer science, a binary tree is a Tree in which each node has at most two child node. Typically the child nodes are called left and right.... data structure
Data structure
A data structure in computer science is a way of storing data in a computer so that it can be used efficiently. It is an organization of mathematical and logical concepts of data....
which has the following properties:
The major advantage of binary search trees over other data structures is that the related sorting algorithm
Sorting algorithm
In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a List in a certain Total order. The most-used orders are numerical order and lexicographical order.... s and search algorithm
Search algorithm
In computer science, a search algorithm, broadly speaking, is an algorithm that takes a problem as input and returns a solution to the problem, usually after evaluating a number of possible solutions.... s such as in-order traversal can be very efficient.
Binary search trees can choose to allow or disallow duplicate values, depending on the implementation.
A data structure in computer science is a way of storing data in a computer so that it can be used efficiently. It is an organization of mathematical and logical concepts of data.... used to construct more abstract data structures such as sets
Set (computer science)
In computer science, a set is a collection of certain values, without any particular Canonical order, and no repeated values. It corresponds with a finite set in mathematics.... , multiset
Multiset
In mathematics, a multiset is a generalization of a Set . A Element of a multiset can have more than one Element , while each member of a set has only one membership.... s, and associative array
Associative array
An associative array is an abstract data type composed of a Collection of unique keys and a collection of values, where each key is associated with one value .... s.
ations on a binary tree require comparisons between nodes.
Computer science is the study of the theoretical foundations of information and computation, and of practical techniques for their implementation and application in computer systems.... , a binary search tree (BST) is a binary tree
Binary tree
In computer science, a binary tree is a Tree in which each node has at most two child node. Typically the child nodes are called left and right.... data structure
Data structure
A data structure in computer science is a way of storing data in a computer so that it can be used efficiently. It is an organization of mathematical and logical concepts of data....
which has the following properties:
Each node (item in the tree) has a distinct value.
Both the left and right subtrees must also be binary search trees.
In computer science, a tree is a widely-used data structure that emulates a hierarchical tree structure with a set of linked Vertex_. It is an acyclic connected graph where each node has a set of zero or more children nodes, and at most one parent node.... of a node contains only values less than the node's value.
The right subtree of a node contains only values greater than the node's value.
The major advantage of binary search trees over other data structures is that the related sorting algorithm
Sorting algorithm
In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a List in a certain Total order. The most-used orders are numerical order and lexicographical order.... s and search algorithm
Search algorithm
In computer science, a search algorithm, broadly speaking, is an algorithm that takes a problem as input and returns a solution to the problem, usually after evaluating a number of possible solutions.... s such as in-order traversal can be very efficient.
Binary search trees can choose to allow or disallow duplicate values, depending on the implementation.
A data structure in computer science is a way of storing data in a computer so that it can be used efficiently. It is an organization of mathematical and logical concepts of data.... used to construct more abstract data structures such as sets
Set (computer science)
In computer science, a set is a collection of certain values, without any particular Canonical order, and no repeated values. It corresponds with a finite set in mathematics.... , multiset
Multiset
In mathematics, a multiset is a generalization of a Set . A Element of a multiset can have more than one Element , while each member of a set has only one membership.... s, and associative array
Associative array
An associative array is an abstract data type composed of a Collection of unique keys and a collection of values, where each key is associated with one value .... s.
Operations
Operations on a binary tree require comparisons between nodes. These comparisons are made with calls to a comparator
Comparator
In electronics, a comparator is a device which compares two voltages or Electric currents and switches its output to indicate which is larger.... , which is a subroutine
Subroutine
In computer science, a subroutine or subprogram is a portion of computer code within a larger computer program, which performs a specific task and is relatively independent of the remaining code.... that computes the total order (linear order) on any two values. This comparator can be explicitly or implicitly defined, depending on the language in which the BST is implemented.
Searching
Searching a binary tree for a specific value can be a recursive
Recursion (computer science)
Recursion is a way of thinking about and solving problems. In fact, Recursion_ is one of the central ideas of computer science. Solving a problem using recursion means the solution depends on solutions to smaller instances of the same problem.... or iterative
Iteration
Iteration means the act of repeating.... process. This explanation covers a recursive method.
In computer science, a tree is a widely-used data structure that emulates a hierarchical tree structure with a set of linked Vertex_. It is an acyclic connected graph where each node has a set of zero or more children nodes, and at most one parent node.... . If the tree is null, the value we are searching for does not exist in the tree. Otherwise, if the value equals the root, the search is successful. If the value is less than the root, search the left subtree. Similarly, if it is greater than the root, search the right subtree. This process is repeated until the value is found or the indicated subtree is null. If the searched value is not found before a null subtree is reached, then the item must not be present in the tree.
Python is a general-purpose high-level programming language. Its design philosophy emphasizes code readability. Python's core syntax and semantics are Minimalism , while the standard library is large and comprehensive.... :
def search_binary_tree(node, key):
if node is None:
return None # key not found
if key < node.key:
return search_binary_tree(node.left, key)
elif key > node.key:
return search_binary_tree(node.right, key)
else: # key is equal to node key
return node.value # found key
This operation requires O
Big O notation
In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.... (log n) time in the average case, but needs O
Big O notation
In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.... (n) time in the worst-case, when the unbalanced tree resembles a linked list
Linked list
In computer science, a linked list is one of the fundamental data structures, and can be used to implement other data structures. It consists of a sequence of node s, each containing arbitrary data Field s and one or two reference s pointing to the next and/or previous nodes.... (degenerate tree
Binary tree
In computer science, a binary tree is a Tree in which each node has at most two child node. Typically the child nodes are called left and right.... ).
Insertion
Insertion begins as a search would begin; if the root is not equal to the value, we search the left or right subtrees as before. Eventually, we will reach an external node and add the value as its right or left child, depending on the node's value. In other words, we examine the root and recursively insert the new node to the left subtree if the new value is less than the root, or the right subtree if the new value is greater than or equal to the root.
Here's how a typical binary search tree insertion might be performed in C++:
/* Inserts the node pointed to by "newNode" into the subtree rooted at "treeNode" */
void InsertNode(Node* &treeNode, Node *newNode)
The above "destructive" procedural variant modifies the tree in place. It uses only constant space, but the previous version of the tree is lost. Alternatively, as in the following Python
Python (programming language)
Python is a general-purpose high-level programming language. Its design philosophy emphasizes code readability. Python's core syntax and semantics are Minimalism , while the standard library is large and comprehensive.... example, we can reconstruct all ancestors of the inserted node; any reference to the original tree root remains valid, making the tree a persistent data structure
Persistent data structure
In computing, a persistent data structure is a data structure which always preserves the previous version of itself when it is modified; such data structures are effectively immutable, as their operations do not update the structure in-place, but instead always yield a new updated structure.... :
def binary_tree_insert(node, key, value):
if node is None:
return TreeNode(None, key, value, None)
The part that is rebuilt uses T(log n) space in the average case and O(n) in the worst case (see big-O notation).
In either version, this operation requires time proportional to the height of the tree in the worst case, which is O
Big O notation
In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.... (log n) time in the average case over all trees, but O(n) time in the worst case.
Another way to explain insertion is that in order to insert a new node in the tree, its value is first compared with the value of the root. If its value is less than the root's, it is then compared with the value of the root's left child. If its value is greater, it is compared with the root's right child. This process continues, until the new node is compared with a leaf node, and then it is added as this node's right or left child, depending on its value.
There are other ways of inserting nodes into a binary tree, but this is the only way of inserting nodes at the leaves and at the same time preserving the BST structure.
Deletion
There are several cases to be considered:
Deleting a leaf: Deleting a node with no children is easy, as we can simply remove it from the tree.
Deleting a node with one child: Delete it and replace it with its child.
Deleting a node with two children: Suppose the node to be deleted is called N. We replace the value of N with either its in-order successor (the left-most child of the right subtree) or the in-order predecessor (the right-most child of the left subtree).
Once we find either the in-order successor or predecessor, swap it with N, and then delete it. Since both the successor and the predecessor must have fewer than two children, either one can be deleted using the previous two cases. A good implementation avoids consistently using one of these nodes, however, because this can unbalance the tree.
C is a general-purpose computer programming language originally developed in 1972 by Dennis Ritchie at the Bell Telephone Laboratories to implement the Unix operating system.... sample code for a recursive version of deletion.
struct Binary_Search_Tree *Bst_Delete(struct Binary_Search_Tree *node, int val)
Although this operation does not always traverse the tree down to a leaf, this is always a possibility; thus in the worst case it requires time proportional to the height of the tree. It does not require more even when the node has two children, since it still follows a single path and does not visit any node twice.
key:
if not (self.leftChild or self.rightChild):
if self
self.parent.leftChild:
self.parent.leftChild = None
else:
self.parent.rightChild = None
elif (self.leftChild or self.rightChild) and (not (self.leftChild and self.rightChild)):
if self.leftChild:
if self
Once the binary search tree has been created, its elements can be retrieved in-order by recursively traversing the left subtree of the root node, accessing the node itself, then recursively traversing the right subtree of the node, continuing this pattern with each node in the tree as it's recursively accessed. The tree may also be traversed in pre-order or post-order traversals.
def traverse_binary_tree(treenode):
if treenode is None: return
left, nodevalue, right = treenode
traverse_binary_tree(left)
visit(nodevalue)
traverse_binary_tree(right)
In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.... time, since it must visit every node. This algorithm is also O(n), and so it is asymptotically optimal
Asymptotically optimal
In computer science, an algorithm is said to be asymptotically optimal if, roughly speaking, for large inputs it performs at worst a constant factor worse than the best possible algorithm.... .
Sort
A binary search tree can be used to implement a simple but efficient sorting algorithm
Sorting algorithm
In computer science and mathematics, a sorting algorithm is an algorithm that puts elements of a List in a certain Total order. The most-used orders are numerical order and lexicographical order.... . Similar to heapsort
Heapsort
Heapsort is a comparison sort sorting algorithm, and is part of the selection sort family. Although somewhat slower in practice on most machines than a good implementation of quicksort, it has the advantage of a worst-case big O notation runtime.... , we insert all the values we wish to sort into a new ordered data structure—in this case a binary search tree—and then traverse it in order, building our result:
def build_binary_tree(values):
tree = None
for v in values:
tree = binary_tree_insert(tree, v)
return tree
def traverse_binary_tree(treenode):
if treenode is None: return []
else:
left, value, right = treenode
return (traverse_binary_tree(left), [value], traverse_binary_tree(right))
The worst-case time of build_binary_tree is —if you feed it a sorted list of values, it chains them into a linked list
Linked list
In computer science, a linked list is one of the fundamental data structures, and can be used to implement other data structures. It consists of a sequence of node s, each containing arbitrary data Field s and one or two reference s pointing to the next and/or previous nodes.... with no left subtrees. For example, build_binary_tree([1, 2, 3, 4, 5]) yields the tree (None, 1, (None, 2, (None, 3, (None, 4, (None, 5, None))))).
There are several schemes for overcoming this flaw with simple binary trees; the most common is the self-balancing binary search tree
Self-balancing binary search tree
In computer science, a self-balancing binary search tree or height-balanced binary search tree is a binary search tree that attempts to keep its height, or the number of levels of nodes beneath the root, as small as possible at all times, automatically.... . If this same procedure is done using such a tree, the overall worst-case time is O
Big O notation
In mathematics, big O notation describes the asymptotic analysis of a function when the argument tends towards a particular value or infinity, usually in terms of simpler functions.... (nlog n), which is asymptotically optimal
Asymptotically optimal
In computer science, an algorithm is said to be asymptotically optimal if, roughly speaking, for large inputs it performs at worst a constant factor worse than the best possible algorithm.... for a comparison sort
Comparison sort
A comparison sort is a type of sorting algorithm that only reads the list elements through a single abstract comparison operation that determines which of two elements should occur first in the final sorted list.... . In practice, the poor cache
CPU cache
A CPU cache is a cache used by the central processing unit of a computer to reduce the average time to access computer storage. The cache is a smaller, faster memory which stores copies of the data from the most frequently used main memory locations.... performance and added overhead in time and space for a tree-based sort (particularly for node allocation
Dynamic memory allocation
In computer science, dynamic memory allocation is the allocation of computer storage storage for use in a computer program during the runtime of that program.... ) make it inferior to other asymptotically optimal sorts such as heapsort
Heapsort
Heapsort is a comparison sort sorting algorithm, and is part of the selection sort family. Although somewhat slower in practice on most machines than a good implementation of quicksort, it has the advantage of a worst-case big O notation runtime.... for static list sorting. On the other hand, it is one of the most efficient methods of incremental sorting, adding items to a list over time while keeping the list sorted at all times...
Example for a Binary Search Tree in Python:
class Node:
def __init__(self, lchild=None, rchild=None, value=-1, data=None):
self.lchild = lchild
self.rchild = rchild
self.value = value
self.data = data
class Bst:
"""Implement Binary Search Tree."""
def __init__(self):
self.l = [] # Nodes
self.root = None
def add(self, key, dt):
"""Add a node in tree."""
if self.root
None:
self.root = Node(value=key, data=dt)
self.l.append(self.root)
return 0
else:
self.p = self.root
while True:
if self.p.value > key:
if self.p.lchild
key:
return -1 # Value already in tree
else:
if self.p.rchild
None:
self.p.rchild = Node(value=key, data=dt)
return 0 # Success
else:
self.p = self.p.rchild
return -2 # Should never happen
def search(self, key):
"""Search Tree for a key and return data; if not found return None."""
self.p = self.root
if self.p
None:
return None
while True:
print self.p.value, self.p.data
if self.p.value > key:
if self.p.lchild
None:
return None # Not found
else:
self.p = self.p.lchild
elif self.p.value
key:
return self.p.data
else:
if self.p.rchild
None:
return None # Not found
else:
self.p = self.p.rchild
return None # Should never happen
def deleteNode(self, key):
"""Delete node with value
key."""
if self.root.value
key:
if self.root.rchild
None:
if self.root.lchild
None:
self.root = None
else: self.root = self.root.lchild
else:
self.root.rchild.lchild = self.root.lchild
self.root = self.root.rchild
return 1
self.p = self.root
while True:
if self.p.value > key:
if self.p.lchild
None:
return 0 # Not found anything to delete
elif self.p.lchild.value
key:
self.p.lchild = self.proceed(self.p, self.p.lchild)
return 1
else:
self.p = self.p.lchild
# There's no way for self.p.value to be equal to key:
if self.p.value < key:
if self.p.rchild
None:
return 0 # Not found anything to delete
elif self.p.rchild.value
There are many types of binary search trees. AVL tree
AVL tree
In computer science, an AVL tree is a self-balancing binary search tree, and it is the first such data structure to be invented. In an AVL tree, the tree height of the two child node subtrees of any node differ by at most one; therefore, it is also said to be height-balanced tree.... s and red-black tree
Red-black tree
A red-black tree is a type of self-balancing binary search tree, a data structure used in computer science, typically used to implement associative arrays.... s are both forms of self-balancing binary search tree
Self-balancing binary search tree
In computer science, a self-balancing binary search tree or height-balanced binary search tree is a binary search tree that attempts to keep its height, or the number of levels of nodes beneath the root, as small as possible at all times, automatically.... s. A splay tree
Splay tree
A splay tree is a self-balancing binary search tree with the additional property that recently accessed elements are quick to access again. It performs basic operations such as insertion, look-up and removal in big O notation amortized analysis time.... is a binary search tree that automatically moves frequently accessed elements nearer to the root. In a treap
Treap
In computer science, a treap is a binary search tree that orders the nodes by adding a priority attribute to a node, as well as a key. The nodes are ordered so that the keys form a binary search tree and the priorities obey the... ("tree heap
Heap (data structure)
In computer science, a heap is a specialized tree data structure-based data structure that satisfies the heap property: if B is a child node of A, then key ≥ key.... "), each node also holds a priority and the parent node has higher priority than its children.
Two other titles describing binary search trees are that of a complete and degenerate tree.
A complete tree is a tree with n levels, where for each level d <= n - 1, the number of existing nodes at level d is equal to 2d. This means all possible nodes exist at these levels. An additional requirement for a complete binary tree is that for the nth level, while every node does not have to exist, the nodes that do exist must fill from left to right.
A degenerate tree is a tree where for each parent node, there is only one associated child node. What this means is that in a performance measurement, the tree will essentially behave like a linked list data structure.
Performance comparisons
D. A. Heger (2004) presented a performance comparison of binary search trees. Treap
Treap
In computer science, a treap is a binary search tree that orders the nodes by adding a priority attribute to a node, as well as a key. The nodes are ordered so that the keys form a binary search tree and the priorities obey the... was found to have the best average performance, while red-black tree
Red-black tree
A red-black tree is a type of self-balancing binary search tree, a data structure used in computer science, typically used to implement associative arrays.... was found to have the smallest amount of performance fluctuations.
Optimal binary search trees
If we don't plan on modifying a search tree, and we know exactly how often each item will be accessed, we can construct an optimal binary search tree, which is a search tree where the average cost of looking up an item (the expected search cost) is minimized.
Assume that we know the elements and that for each element, we know the proportion of future lookups which will be looking for that element. We can then use a dynamic programming
Dynamic programming
In mathematics and computer science, dynamic programming is a method of solving problems that exhibit the properties of overlapping subproblems and optimal substructure .... solution, detailed in section 15.5 of Introduction to Algorithms (Second Edition) by Thomas H. Cormen, to construct the tree with the least possible expected search cost.
Even if we only have estimates of the search costs, such a system can considerably speed up lookups on average. For example, if you have a BST of English words used in a spell checker
Spell checker
In computing, a spell checker is an application software that flags words in a document that may not be spelling correctly. Spell checkers may be stand-alone capable of operating on a block of text, or as part of a larger application, such as a word processor, email client, electronic dictionary, or search engine.... , you might balance the tree based on word frequency in text corpora
Text corpus
In linguistics, a corpus or text corpus is a large and structured set of texts . They are used to do statistical analysis and hypothesis testing, checking occurrences or validating linguistic rules on a specific universe.... , placing words like "the" near the root and words like "agerasia" near the leaves. Such a tree might be compared with Huffman trees, which similarly seek to place frequently-used items near the root in order to produce a dense information encoding; however, Huffman trees only store data elements in leaves and these elements need not be ordered.
If we do not know the sequence in which the elements in the tree will be accessed in advance, we can use splay tree
Splay tree
A splay tree is a self-balancing binary search tree with the additional property that recently accessed elements are quick to access again. It performs basic operations such as insertion, look-up and removal in big O notation amortized analysis time.... s which are asymptotically as good as any static search tree we can construct for any particular sequence of lookup operations.
Alphabetic trees are Huffman trees with the additional constraint on order, or, equivalently, search trees with the modification that all elements are stored in the leaves. Faster algorithms exist for optimal alphabetic binary trees (OABTs).
Example:
procedure Optimum Search Tree(f, f´, c)
for j = 0 to n do
c[j, j] = 0, F[j, j] = f´j
for d = 1 to n do
for i = 0 to (n - d) do
j = i + d
F[i, j] = F[i, j - 1] + f´ + f´j
c[i, j] = MIN(i
Donald Ervin Knuth is a renowned computer science and Emeritus of the Art of Computer Programming at Stanford University.Author of the seminal multi-volume work The Art of Computer Programming , Knuth has been called the "father" of the run-time analysis, contributing to the development of, and systematizing formal mathematical techn... . The Art of Computer Programming, Volume 3: Sorting and Searching, Third Edition. Addison-Wesley, 1997. ISBN 0-201-89685-0. Section 6.2.2: Binary Tree Searching, pp.426–458.
Thomas H. Cormen is the co-author of Introduction to Algorithms, along with Charles Leiserson, Ron Rivest, and Clifford Stein. He is a Full Professor of computer science at Dartmouth College and currently Chair of the Dartmouth College Writing Program.... , Charles E. Leiserson
Charles E. Leiserson
Charles Eric Leiserson is a computer scientist, specializing in the theory of parallel computing and distributed computing, and particularly practical applications thereof; as part of this effort, he developed the Cilk multithreaded language.... , Ronald L. Rivest, and Clifford Stein
Clifford Stein
Clifford Stein, a computer scientist, is currently a professor of industrial engineering and operations research at Columbia University in New York, NY, where he also holds an appointment in the Department of Computer Science.... . Introduction to Algorithms
Introduction to Algorithms
Introduction to Algorithms is a book by Thomas H. Cormen, Charles E. Leiserson, Ron Rivest, and Clifford Stein. It is used as the textbook for algorithms courses at many universities.... , Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0-262-03293-7. Chapter 12: Binary search trees, pp.253–272. Section 15.5: Optimal binary search trees, pp.356–363.
