Binary search tree
In computer science,[] a binary search tree, also called an ordered or sorted binary tree, is a rooted binary tree whose internal nodes each store a key greater than all the keys in the node's left subtree and less than those in its right subtree. A binary tree is a type of data structure for storing data such as numbers in an organized way. Binary search trees allow binary search for fast lookup, addition and removal of data items, and can be used to implement dynamic sets and lookup tables. The order of nodes in a BST means that each comparison skips about half of the remaining tree, so the whole lookup takes time proportional to the binary logarithm of the number of items stored in the tree. This is much better than the linear time required to find items by key in an array, but slower than the corresponding operations on hash tables. Several variants of the binary search tree have been studied.
Definition
A binary search tree is a rooted binary tree, whose internal nodes each store a key, and each has two distinguished sub-trees, commonly denoted left and right. The tree additionally satisfies the binary search property: the key in each node is greater than or equal to any key stored in the left sub-tree, and less than or equal to any key stored in the right sub-tree. The leaves of the tree contain no key and have no structure to distinguish them from one another.Often, the information represented by each node is a record rather than a single data element. However, for sequencing purposes, nodes are compared according to their keys rather than any part of their associated records. The major advantage of binary search trees over other data structures is that the related sorting algorithms and search algorithms such as in-order traversal can be very efficient; they are also easy to code.
Binary search trees are a fundamental data structure used to construct more abstract data structures such as sets, multisets, and associative arrays.
- When inserting or searching for an element in a binary search tree, the key of each visited node has to be compared with the key of the element to be inserted or found.
- The shape of the binary search tree depends entirely on the order of insertions and deletions and can become degenerate.
- After a long intermixed sequence of random insertion and deletion, the expected height of the tree approaches square root of the number of keys,, which grows much faster than.
- There has been a lot of research to prevent degeneration of the tree resulting in worst case time complexity of .
Order relation
For a search function supporting and handling duplicates in a tree, see section [|Searching with duplicates allowed].
In the context of binary search trees, a total preorder is realized most flexibly by means of a three-way comparison subroutine.
Operations
Binary search trees support three main operations: insertion of elements, deletion of elements, and lookup.Searching
Searching in a binary search tree for a specific key can be programmed recursively or iteratively.We begin by examining the root node. If the tree is null, the key we are searching for does not exist in the tree. Otherwise, if the key equals that of the root, the search is successful and we return the node. If the key is less than that of the root, we search the left subtree. Similarly, if the key is greater than that of the root, we search the right subtree. This process is repeated until the key is found or the remaining subtree is null. If the searched key is not found after a null subtree is reached, then the key is not present in the tree. This is easily expressed as a recursive algorithm :
def search_recursively:
if node None or node.key key:
return node
if key < node.key:
return search_recursively
if key > node.key:
return search_recursively
The same algorithm can be implemented iteratively:
def search_iteratively:
current_node = node
while current_node != None:
if key current_node.key:
return current_node
if key < current_node.key:
current_node = current_node.left
else: # key > current_node.key:
current_node = current_node.right
return current_node
These two examples do not support duplicates, i. o. w. they rely on the order relation being a total order.
One can note that the recursive algorithm is tail recursive. In a language supporting tail call optimization, the recursive and iterative examples will compile to equivalent programs.
Because in the worst case this algorithm must search from the root of the tree to the leaf farthest from the root, the search operation takes time proportional to the tree's height. On average, binary search trees with keys have height. However, in the worst case, binary search trees can have height, when the unbalanced tree resembles a linked list.
Searching with duplicates allowed
If the order relation is only a total preorder, a reasonable extension of the functionality is the following: also in case of equality search down to the leaves. Thereby allowing to specify a direction, where to insert a duplicate, either to the right or left of all duplicates in the tree so far. If the direction is hard-wired, both choices, right and left, support a stack with insert duplicate as push operation and delete as the pop operation.def search_duplicatesAllowed:
current_node = node
while current_node != None:
if key < current_node.key:
current_node = current_node.left
else: # key >= current_node.key:
current_node = current_node.right
return current_node
A binary tree sort equipped with such a search function becomes stable.
Insertion
Insertion begins as a search would begin; if the key is not equal to that of the root, we search the left or right subtrees as before. Eventually, we will reach an external node and add the new key-value pair as its right or left child, depending on the node's key. In other words, we examine the root and recursively insert the new node to the left subtree if its key is less than that of the root, or the right subtree if its key is greater than or equal to the root.Here's how a typical binary search tree insertion might be performed in a binary tree in C++:
void insert
Alternatively, a non-recursive version might be implemented like this. Using a pointer-to-pointer to keep track of where we came from lets the code avoid explicit checking for and handling of the case where it needs to insert a node at the tree root:
void insert
The above destructive procedural variant modifies the tree in place. It uses only constant heap space, but the prior version of the tree is lost. Alternatively, as in the following Python 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:
def binary_tree_insert:
if node None:
return NodeTree
if key node.key:
return NodeTree
if key < node.key:
return NodeTree
return NodeTree
The part that is rebuilt uses space in the average case and in the worst case.
In either version, this operation requires time proportional to the height of the tree in the worst case, which is time in the average case over all trees, but time in the worst case.
Another way to explain insertion is that in order to insert a new node in the tree, its key is first compared with that of the root. If its key is less than the root's, it is then compared with the key of the root's left child. If its key 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 key: if the key is less than the leaf's key, then it is inserted as the leaf's left child, otherwise as the leaf's right child.
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
When removing a node from a binary search tree it is mandatory to maintain the in-order sequence of the nodes.There are many possibilities to do this. However, the following method which has been proposed by T. Hibbard in 1962 guarantees that the heights of the subject subtrees are changed by at most one.
There are three possible cases to consider:
- Deleting a node with no children: simply remove the node from the tree.
- Deleting a node with one child: remove the node and replace it with its child.
- Deleting a node with two children: call the node to be deleted D. Do not delete D. Instead, choose either its in-order predecessor node or its in-order successor node as replacement node E. Copy the user values of E to D. If E does not have a child simply remove E from its previous parent G. If E has a child, say F, it is a right child. Replace E with F at E
's parent.
Nodes with two children are harder to delete. A node's in-order successor is its right subtree's left-most child, and a node's in-order predecessor is the left subtree's right-most child. In either case, this node will have only one or no child at all. Delete it according to one of the two simpler cases above.
Consistently using the in-order successor or the in-order predecessor for every instance of the two-child case can lead to an unbalanced tree, so some implementations select one or the other at different times.
Runtime analysis: 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.
def find_min:
"""Get minimum node in a subtree."""
current_node = self
while current_node.left_child:
current_node = current_node.left_child
return current_node
def replace_node_in_parent -> None:
if self.parent:
if self self.parent.left_child:
self.parent.left_child = new_value
else:
self.parent.right_child = new_value
if new_value:
new_value.parent = self.parent
def binary_tree_delete -> None:
if key < self.key:
self.left_child.binary_tree_delete
return
if key > self.key:
self.right_child.binary_tree_delete
return
# Delete the key here
if self.left_child and self.right_child: # If both children are present
successor = self.right_child.find_min
self.key = successor.key
successor.binary_tree_delete
elif self.left_child: # If the node has only a *left* child
self.replace_node_in_parent
elif self.right_child: # If the node has only a *right* child
self.replace_node_in_parent
else:
self.replace_node_in_parent # This node has no children
Traversal
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. As with all binary trees, one may conduct a pre-order traversal or a post-order traversal, but neither are likely to be useful for binary search trees. An in-order traversal of a binary search tree will always result in a sorted list of node items.The code for in-order traversal in Python is given below. It will call callback for every node in the tree.
def traverse_binary_tree:
if node None:
return
traverse_binary_tree
callback
traverse_binary_tree
Traversal requires time, since it must visit every node. This algorithm is also, so it is asymptotically optimal.
Traversal can also be implemented iteratively. For certain applications, e.g. greater equal search, approximative search, an operation for single step traversal can be very useful. This is, of course, implemented without the callback construct and takes on average and in the worst case.
Verification
Sometimes we already have a binary tree, and we need to determine whether it is a BST. This problem has a simple recursive solution.The BST property—every node on the right subtree has to be larger than the current node and every node on the left subtree has to be smaller than the current node—is the key to figuring out whether a tree is a BST or not. The greedy algorithm—simply traverse the tree, at every node check whether the node contains a value larger than the value at the left child and smaller than the value on the right child—does not work for all cases. Consider the following tree:
20
/ \
10 30
/ \
5 40
In the tree above, each node meets the condition that the node contains a value larger than its left child and smaller than its right child hold, and yet it is not a BST: the value 5 is on the right subtree of the node containing 20, a violation of the BST property.
Instead of making a decision based solely on the values of a node and its children, we also need information flowing down from the parent as well. In the case of the tree above, if we could remember about the node containing the value 20, we would see that the node with value 5 is violating the BST property contract.
So the condition we need to check at each node is:
- if the node is the left child of its parent, then it must be smaller than the parent and it must pass down the value from its parent to its right subtree to make sure none of the nodes in that subtree is greater than the parent
- if the node is the right child of its parent, then it must be larger than the parent and it must pass down the value from its parent to its left subtree to make sure none of the nodes in that subtree is lesser than the parent.
struct TreeNode ;
bool isBST
node->key+1 and node->key-1 are done to allow only distinct elements in BST.If we want the same elements to also be present, then we can use only
node->key in both places.The initial call to this function can be something like this:
if else
Essentially we keep creating a valid range and keep shrinking it down for each node as we go down recursively.
As pointed out in section #Traversal, an in-order traversal of a binary search tree returns the nodes sorted. Thus we only need to keep the last visited node while traversing the tree and check whether its key is smaller compared to the current key.
Examples of applications
Sort
A binary search tree can be used to implement a simple sorting algorithm. Similar to heapsort, 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.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 with no left subtrees. For example, build_binary_tree yields the tree )).There are several schemes for overcoming this flaw with simple binary trees; the most common is the self-balancing binary search tree. If this same procedure is done using such a tree, the overall worst-case time is, which is asymptotically optimal for a comparison sort. In practice, the added overhead in time and space for a tree-based sort make it inferior to other asymptotically optimal sorts such as heapsort 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.
Priority queue operations
Binary search trees can serve as priority queues: structures that allow insertion of arbitrary key as well as lookup and deletion of the minimum key. Insertion works as previously explained. Find-min walks the tree, following left pointers as far as it can without hitting a leaf:// Precondition: T is not a leaf
function find-min:
while hasLeft:
T ? left
return key
Find-max is analogous: follow right pointers as far as possible. Delete-min can simply look up the minimum, then delete it. This way, insertion and deletion both take logarithmic time, just as they do in a binary heap, but unlike a binary heap and most other priority queue implementations, a single tree can support all of find-min, find-max, delete-min and delete-max at the same time, making binary search trees suitable as double-ended priority queues.
Types
There are many types of binary search trees. AVL trees and red-black trees are both forms of self-balancing binary search trees. A splay tree is a binary search tree that automatically moves frequently accessed elements nearer to the root. In a treap, each node also holds a priority and the parent node has higher priority than its children. Tango trees are trees optimized for fast searches.T-trees are binary search trees optimized to reduce storage space overhead, widely used for in-memory databases
A degenerate tree is a tree where for each parent node, there is only one associated child node. It is unbalanced and, in the worst case, performance degrades to that of a linked list. If your add node function does not handle re-balancing, then you can easily construct a degenerate tree by feeding it with data that is already sorted. 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 presented a performance comparison of binary search trees. Treap was found to have the best average performance, while red-black tree was found to have the smallest number of performance variations.Optimal binary search trees
If we do not 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 is minimized.Even if we only have estimates of the search costs, such a system can considerably speed up lookups on average. For example, if we have a BST of English words used in a spell checker, we might balance the tree based on word frequency in text corpora, 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 store data elements only in leaves, and these elements need not be ordered.
If the sequence in which the elements in the tree will be accessed is unknown in advance, splay trees can be used 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.