Database uses binary search tree
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 implemented in Python:. If the order relation is only a total preorder a reasonable extension of the functionality is the following: A binary tree sort equipped with such a comparison function becomes stable.
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 see tree terminology. On average, binary search trees with n nodes have O log n height.
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 here encoded as a record 'newNode' 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.
The above destructive procedural variant modifies the tree in place. It uses only constant heap space and the iterative version uses constant stack space as well , 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:. The part that is rebuilt uses O log n space in the average case and O n in the worst case.
In either version, this operation requires time proportional to the height of the tree in the worst case, which is O 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 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: 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. 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  guarantees that the heights of the subject subtrees are changed by at most one.
There are three possible cases to consider:. Broadly speaking, nodes with children are harder to delete. As with all binary trees, 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.
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.
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 numbers, strings or other comparable items. The code for in-order traversal in Python is given below. It will call callback some function the programmer wishes to call on the node's value, such as printing to the screen for every node in the tree.
Traversal requires O n time, since it must visit every node. This algorithm is also O n , so it is asymptotically optimal. Traversal can also be implemented iteratively. For certain applications, e. This is, of course, implemented without the callback construct and takes O 1 on average and O log n in the worst case.
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:. 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: 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.
As pointed out in section Traversal , an in-order traversal of a binary search tree returns the nodes sorted. 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.
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 O n log n , which is asymptotically optimal for a comparison sort. In practice, the added overhead in time and space for a tree-based sort particularly for node allocation make it inferior to other asymptotically optimal sorts such as heapsort for static list sorting.
The search algorithm is simple, but it does not minimize the number of database accesses required to reach a desired record. When the entire tree is contained in RAM, which is a fast-read, fast-write medium, the number of required accesses is of little concern. But when some or all of the data is on disk, which is slow-read, slow-write, it is advantageous to minimize the number of accesses the tree depth.
Alternative algorithms such as the B-tree accomplish this. Also see binary search and tree structure. Compare B-tree, M-tree, splay tree , and X-tree. Cloud-based database software adds new IT options for users. By submitting you agree to receive email from TechTarget and its partners. If you reside outside of the United States, you consent to having your personal data transferred to and processed in the United States.
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