Binary search tree. Lookup operation

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An important special kind of binary tree is the binary search tree BST. In a BST, each node stores some information including a unique key value and perhaps some associated data.

A binary tree is a BST iff, for every node nin the tree:. In these notes, we will assume that duplicates are not allowed. Note that more than one BST can be used to store the same set of key values. For example, both of the following are BSTs that store the same set of integer keys:. The reason binary-search trees are important is that the following operations can be implemented efficiently using a BST:.

Which of the following binary trees are BSTs? If a tree is not a BST, say why. Using which kind of traversal pre-order, post-order, in-order, or level-order visits the simple binary search tree implementation in java of a BST in sorted order? To implement a binary search tree, we will use two classes: The following class definitions assume that the BST will store only key values, no associated data. The type parameter K is the type of the key.

To implement a BST that stores some data with each key, we would use the following class definitions changes are in red:. From now on, we will assume that BSTs only store key values, not associated data. We will also assume that null is not a valid key value i. In general, to determine whether a given value is in the BST, we will start at the root of the tree and determine whether the value we are looking for:.

If neither base case holds, a recursive lookup is done on the appropriate subtree. Since all values less than the root's value are in the left subtree and all values greater than the root's value are in the right subtree, there is no point in looking in both subtrees: The code for the lookup method uses an auxiliary, recursive method with the same name i. How much time does it take to search for a value in a BST? Note that lookup always follows a path from the root down towards a leaf.

In the worst case, it goes all the way to a leaf. Therefore, the worst-case time is proportional to the length of the longest path from the root to a leaf the height of the tree. In general, we'd like to know how much time is required for lookup as simple binary search tree implementation in java function of the number of values stored in the tree.

In other words, what is the relationship between the number of nodes in a BST and the height of the tree? This depends on the "shape" of the tree. In the worst case, all nodes have just one child, and the tree is essentially a linked list. Searching for values in the range and will require following the path from the root down to the leaf the node containing the value 20i.

In the best case, all nodes have 2 children and all leaves are at the same depth, for example:. In general, a tree like this a full tree will have height approximately log 2 Nwhere N is the number of nodes in the tree.

The value log 2 N is roughly the number of times you can divide N by two before you get to zero. The reason we use log 2. However, when we use big-O notation, we just say that the height of a full tree with N nodes is O log N -- we drop simple binary search tree implementation in java "2" subscript, because log 2 N is proportional to log k N for any constant k, i. In the worst case a "linear" tree this is O Nwhere N is the number of nodes in the tree. In the best case a "full" tree this is O log Simple binary search tree implementation in java.

Where should a new item go in a BST? The answer is easy: If you don't put it there then you won't find it later. Here are pictures illustrating what happens when we insert the value 15 into the example tree used above. It is easy to see that the complexity for insert is the same as for lookup: As mentioned above, the order in which values are inserted determines what BST is built inserting the same values in different orders can result in different final BSTs.

Draw the BST that results from inserting the values 1 to 7 in each of the following orders reading from left to right:. As you would expect, deleting an item involves a search to locate the node that contains the value to be deleted.

Here is an outline of the code for the delete method. If the search for the node containing the value to be deleted succeeds, there are three cases to deal with:. When the node to delete is a leaf, we want to remove it from the BST by setting the appropriate child pointer of its parent to null or by setting root to null if the node to be deleted is the root and it has no children.

Note that the call to delete was one of the following:. So in all three cases, the right thing happens if the delete method just returns null. When the node to delete has one child, we can simply replace that node with its child by returning a pointer to that child.

As an example, let's delete 16 from the BST just formed:. Here's the code for deletehandling the two cases we've discussed so far the new code is shown in red:.

The hard case is simple binary search tree implementation in java the node to delete has two children. We'll call the node to delete n. We can't replace node n with one of its children, because what would we do with the other child? Instead, we will replace the key in node n with the key value v from another node lower down in the tree, then recursively delete value v.

The question is what value can we use simple binary search tree implementation in java replace n 's key? We have to choose that value so that the tree is still a BST, i.

There are two possibilities that work: We'll arbitrarily decide to use the smallest value in the right subtree. To find that value, we just follow a path in the right subtree, always going to the left child, since smaller values are in left subtrees. Once the value is simple binary search tree implementation in java, we copy it into simple binary search tree implementation in java nthen we recursively delete that value from n 's right subtree.

Here's the final version of the delete method:. Below is a slightly different example BST; let's see what happens when we delete 13 from that tree. Write the auxiliary method smallest used by the delete method given above. The header for smallest is:. If the node to be deleted has zero or one child, then the delete method will "follow a path" from the root to that node. So the worst-case time is proportional to the height of the tree just like for lookup and insert.

So in the worst case, a path from the root to a leaf is followed twice. Since we don't care about constant simple binary search tree implementation in java, the time is still proportional to the height of the tree. The Java standard library has built into it an industrial-strength version of binary search trees, so if you are programming in Java and need this functionality, you would be better off using the library version rather than writing your own.

The class that most closely matches the outline above, in which the nodes contain only keys and no other data, is called TreeSet. Class TreeSet is an implementation of the Set interface. There simple binary search tree implementation in java another implementation, called HashSetthat we will study later in this course.

Here's an example of how you might use a Set to implement a simple spell-checker. This simple binary search tree implementation in java used a set of String. You could also have a set of Integera set of Floator a set of any other type of object, so long as the type implements Comparable. If you want to associate data with each key, use interface Map and the corresponding class TreeMap.

For example, if you want to quickly look up an Employee given his employee number, you should use a Map rather than a Set to keep track of employees. As another example, here is a complete program that counts the number of occurrences of each word in a document. Without it, the program would look for "words" separated by spaces, considering "details" and "details. See the documentation for Scanner and Pattern for more details. The value type V can be any class or interface. The key type K can be any class or interface that implements Comparablefor example.

The method put key, value returns the value previously associated with key if any or null if key is a new key. The method get key returns the value associated with key or null if there is no such value. Both keys and values can be null. If your program stores null values in the map, you should use containsKey key to check whether a particular key is present. Map has many other useful methods. Of particular note are sizeremove keyclearand keySet.

The keySet method returns a Set containing all the keys currently in the map. The CountWords example uses it to list the words in the document. A binary search tree can be used to store any objects that implement the Comparable interface i. A BST can also be used to store Comparable objects simple binary search tree implementation in java some associated data. The advantage of using a binary search tree instead of, say, a linked list is that, if the tree is reasonably balanced shaped more like a "full" tree than like a "linear" treethe insertlookupand delete operations can all be implemented to simple binary search tree implementation in java in O log N time, where N is the number of stored items.

For a linked list, although insert can be implemented to run in O 1 time, lookup and delete take O N time in the worst case. Logarithmic time is generally much faster than linear time.

Of course, it is important to remember that for a "linear" tree one in which every node has one childthe worst-case times for insertlookupand delete will be O N.

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In computer science , binary search trees BST , sometimes called ordered or sorted binary trees , are a particular type of container: They allow fast lookup, addition and removal of items, and can be used to implement either dynamic sets of items, or lookup tables that allow finding an item by its key e.

Binary search trees keep their keys in sorted order, so that lookup and other operations can use the principle of binary search: On average, this means that each comparison allows the operations to skip about half of the tree, so that each lookup, insertion or deletion takes time proportional to the 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 unsorted array, but slower than the corresponding operations on hash tables.

Several variants of the binary search tree have been studied in computer science; this article deals primarily with the basic type, making references to more advanced types when appropriate.

A binary search tree is a rooted binary tree , whose internal nodes each store a key and optionally, an associated value and each have two distinguished sub-trees, commonly denoted left and right. The tree additionally satisfies the binary search property, which states that the key in each node must be 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.

Frequently, 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. Binary search requires an order relation by which every element item can be compared with every other element in the sense of a total preorder.

The part of the element which effectively takes place in the comparison is called its key. In the context of binary search trees a total preorder is realized most flexibly by means of a three-way comparison subroutine.

Binary search trees support three main operations: Searching 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 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 [2] 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.

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.

Binary search trees can serve as priority queues: Insertion works as previously explained. Find-min walks the tree, following left pointers as far as it can without hitting a leaf:. Delete-min max can simply look up the minimum maximum , 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.

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 tree heap , each node also holds a randomly chosen 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.

Heger [4] 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 amount of performance variations. If we do not plan on modifying a search tree, and we know exactly how often each item will be accessed, we can construct [5] 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.

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 , you 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.