Binary Search Tree

The advantages of the tree data structure over linked list data structure are many. In a linked list, a programmer has to search the whole list to find out a duplicate of a number to be inserted. It is very tedious job as the number of stored items in a linked list is very large. But in case of tree data structure, we get a dynamic structure in which any number of items as long as memory is available, can be stored. By using tree data structure, the search operation can be carried out very fast. Now we will see how the use of binary tree can help in searching the duplicate number in a very fast manner.

 

Cost of Search

Consider the previous example where we inserted the number 17 in the tree. We executed a while loop in the insert method and carried out a comparison in while loop. If the comparison is true, it will reflect that in this case, the number in the node where the pointer p is pointing is not equal to 17 and also q is not NULL. Then we move p actually q to the left or right side. This means that if the condition of the while loop is true then we go one level down in the tree. Thus we can understand it easily that if there is a tree of 6 levels, the while loop will execute maximum 6 times. We conclude from it that in a given binary tree of depth d, the maximum number of executions of the while loop will be equal to d. The code after the while loop will do the process depending upon the result of the while loop. It will insert the new number or display a message if the number was already there in the tree.

Now suppose we have another method find. This method does not insert a new number in the tree. Rather, it traverses a tree to find if a given number is already present in the tree or not. The tree which the find method traverses was made in such an order that all the numbers less than the number at the root are in the left sub-tree of the root while the right sub-tree of the root contains the numbers greater than the number at the root. Now the find method takes a number x and searches out its duplicate in the given tree. The find method will return true if x is present in the tree. Otherwise, it will return false. This find method does the same process that a part of the insert method performs. The difference is that the insert method checks for a duplicate number before putting a number in the tree whereas the find method only finds a number in the tree. Here in the find method, the while loop is also executed at maximum equal to the number of levels of the tree. We do a comparison at each level of the tree until either x is found or q becomes NULL. The loop terminates in case, the number is found or it executes to its maximum number, i.e. equal to the number of levels of the tree.

In the discussion on binary tree, we talked about the level and number of nodes of a binary tree. It was witnessed that if we have a complete binary tree with n numbers of nodes, the depth d of the tree can be found by the following equation:

d = log₂ (n + 1) – 1

Suppose we have a complete binary tree in which there are 100,000 nodes, then its depth d will be calculated in the following fashion.

d = log₂ (100000 + 1) – 1 = log₂ (100001) – 1= 20

The statement shows that if there are 100,000 unique numbers (nodes) stored in a complete binary tree, the tree will have 20 levels. Now if we want to find a number x in this tree (in other words, the number is not in the tree), we have to make maximum 20 comparisons i.e. one comparison at each level. Now we can understand the benefit of tree as compared to the linked list. If we have a linked list of 100,000 numbers, then there may be 100,000 comparisons (supposing the number is not there) to find a number x in the list.

Thus in a tree, the search is very fast as compared to the linked list. If the tree is complete binary or near-to-complete, searching through 100,000 numbers will require a maximum of 20 comparisons or in general, approximately log₂(n). Whereas in a linked list, the comparisons required could be a maximum of n.

Tree with the linked structure, is not a difficult data structure. We have used it to allocate memory, link and set pointers to it. It is not much difficult process. In a tree, we link the nodes in such a way that it does not remain a linear structure. If instead of 100,000, we have 1 million or 10 million or say, 1 billion numbers and want to build a complete binary tree of these numbers, the depth (number of levels) of the tree will be log₂ of these numbers. The log₂ of these numbers will be a small number, suppose 25, 30 or 40. Thus we see that the number of level does not increase in such a ratio as the number of nodes increase. So we can search a number x in a complete binary tree of 1 billion nodes only in 30-40 comparisons. As the linked list of such a large number grows large, the search of a number in such a case will also get time consuming process. The usage of memory space does not cause any effect in the linked list and tree data structures. We use the memory dynamically in both structures. However, time is a major factor. Suppose one comparison takes one micro second, then one billion seconds are required to find a number from a linked list (we are supposing the worst case of search where traversing of the whole list may be needed). This time will be in hours. On the other hand, in case of building a complete binary tree of these one billion numbers, we have to make 30-40 comparisons (as the levels of the tree will be 30-40), taking only 30-40 microseconds. We can clearly see the difference between hours and microseconds. Thus it is better to prefer the process of building a tree of the data to storing it in a linked list to make the search process faster.

 

Binary Search Tree

While discussing the search procedure, the tree for search was built in a specific order. The order was such that on the addition of a number in the tree, we compare it with a node. If it is less than this, it can be added to the left sub-tree of the node. Otherwise, it will be added on the right sub-tree. This way, the tree built by us has numbers less than the root in the left sub-tree and the numbers greater than the root in the right sub-tree. A binary tree with such a property that items in the left sub-tree are smaller than the root and items in the right sub-tree are larger than the root is called a binary search tree (BST). The searching and sorting operations are very common in computer science. We will be discussing them many times during this course. In most of the cases, we sort the data before a search operation. The building process of a binary search tree is actually a process of storing the data in a sorted form. The BST has many variations, which will be discussed later. The BST and its variations play an important role in searching algorithms. As data in a BST is in an order, it may also be termed as ordered tree.

 

Traversing a Binary Tree

Now let’s discuss the ways to print the numbers present in a BST. In a linked list, the printing of stored values is easy. It is due to the fact that we know wherefrom, a programmer needs to start and where the next element is. Equally is true about printing of the elements in an array. We execute a for loop starting from the first element (i.e. index 0) to the last element of the array. Now let’s see how we can traverse a tree to print (display) the numbers (or any data items) of the tree.

We can explain this process with the help of the following example in which we traverse a binary search tree. Suppose there are three nodes tree with three numbers stored in it as shown below.

Fig 13.1: A three node binary tree

Here we see that the root node has the number 14. The left sub-tree has only one node i.e. number 4. Similarly the right sub-tree consists of a single node with the number 15. If we apply the permutations combinations on these three nodes to print them, there may be the following six possibilities.

1: (4, 14, 15)

2: (14, 4, 15)

3: (15, 4, 14)

4: (4, 15, 14)

5: (14, 15, 4)

6: (15, 14, 4)

Look at these six combinations of printing the nodes of the tree. In the first combination, the order of printing the nodes is 4-14-15. It means that left subtree-root-right subtree. In the second combination the order is root-left subtree-right subtree. In the third combination, the order of printing the nodes is right subtree-root-left subtree. The fourth combination has left subtree-right subtree-root. The fifth combination has the order root-rigth subtree- left subtree. Finally the sixth combination has the order of printing the nodes right subtree-root-left subtree. These six possibilities are for the three nodes only. If we have a tree having a large number of nodes, then there may increase number of permutations for printing the nodes.

Let’s see the general procedure of traversing a binary tree. We know by definition that a binary tree consists of three sets i.e. root, left subtree and right subtree. The following figure depicts a general binary tree.

Fig 13.2: A generic binary tree

In this figure, we label the root node with N. The left subtree in the figure is in a triangle labeled as L. This left subtree may consist of any number of nodes. Similarly the right subtree is enclosed in a triangle having the label R. This triangle of right subtree may contain any number of nodes. Following are the six permutations, which we have made for the three nodes previously. To generalize these permutations, we use N, L and R as abbreviations for root, left subtree and right subtree respectively.

1: (L, N, R)

2: (N, L, R)

3: (R, L, N)

4: (L, R, N)

5: (N, R, L)

6: (R, N, L)

In these permutations, the left and right subtree are not single nodes. These may consist of several nodes. Thus where we see L in the permutations, it means traversing the left subtree. Similarly R means traversing the right subtree. In the previous tree of three nodes, these left and right subtrees are of single nodes. However, they can consist of any number of nodes. We select the following three permutations from the above six. The first of these three is (N, L, R), also called as preorder traversal. The second permutation is (L, N, R) which we called inorder traversal. Finally the third permutation, also termed as postorder traversal is (L, R, N). Now we will discuss these preorder, inorder and postorder traversal in detail besides having a look on their working. We will also see the order in which the numbers in the tree are displayed by these traversing methods.

C++ code

Let’s write the C++ code for it. Following is the code of the preorder method.

void preorder(TreeNode<int>* treeNode) { if( treeNode != NULL ) { cout << *(treeNode->getInfo())<<" "; preorder(treeNode->getLeft()); preorder(treeNode->getRight()); } }

In the arguments, there is a pointer to a TreeNode. We may start from any node and the pointer of the node will be provided as argument to the preorder method. In this method, first of all we check whether the pointer provided is NULL or not. If it is not NULL, we print the information stored in that node with the help of the getInfo() method. Then we call the getLeft() method that returns a pointer of left node, which may be a complete subtree. With the help of this method, we get the root of that subtree. We call the preorder method again passing that pointer. When we return from that, the preorder method is called for the right node. Let’s see what is happening in this method. We are calling the preorder method within the preorder method. This is actually a recursive call. Recursion is supported in C++ and other languages. Recursion means that a function can call itself. We may want to know why we are doing this recursive call. We will see some more examples in this regard and understand the benefits of recursive calls. For the time being, just think that we are provided with a tree with a root pointer. In the preorder, we have to print the value of the root node. Don’t think that you are in the preorder method. Rather keep in mind that you have a preorder function. Suppose you want to print out the left subtree in the preorder way. For this purpose, we will call the preorder function. When we come back, the right subtree will be printed. In the right subtree, we will again call the preorder function that will print the information. Then call the preorder function for the left subtree and after that its right subtree. It will be difficult if you try to do this incursion in the mind. Write the code and execute it. You must be knowing that the definition of the binary tree is recursive. We have a node and left and right subtrees. What is left subtree? It is also a node with a left subtree and right subtree. We have shown you how the left subtree and right subtree are combined to become a tree. The definition of tree is itself recursive. You have already seen the recursive functions. You have seen the factorial example. What is factorial of N? It is N multiplied by N-1 factorial. What is N-1 factorial? N-1 factorial is N-1 multiplied by N-2 factorial. This is the recursive definition. For having an answer, it is good to calculate the factorial of one less number till the time you reach at the number 2. You will see these recursions or recursive relations here and also in mathematic courses. In the course of discrete mathematics, recursion method is used. Here we are talking about the recursive calls. We will now see an example to understand how this recursive call works and how can we traverse the tree using the recursion. One of the benefits of recursion that it prints out the information of all the nodes without caring for the size of the tree. If the tree has one lakh nodes, this simple four lines routine will print all the nodes. When compared with the array printing, we have a simple loop there. In the link list also, we have a small loop that executes till the time we have a next pointer as NULL. For tree, we use recursive calls to print it.

Here is the code of the inorder function.

void inorder(TreeNode<int>* treeNode) { if( treeNode != NULL ) { inorder(treeNode->getLeft()); cout << *(treeNode->getInfo())<<" "; inorder(treeNode->getRight()); } }

The argument is the same as in the preorder i.e. a pointer to the TreeNode. If this node is not NULL, we call getLeft() to get the left node and call the inorder function. We did not print the node first here. In the inorder method, there is no need to print the root tree first of all. We start with the left tree. After completely traversing the complete left tree, we print the value of the node. In the end, we traverse the right subtree in recursion.

Hopefully, you have now a fair knowledge about the postorder mechanism too. Here is the code of the postorder method.

void postorder(TreeNode<int>* treeNode) { if( treeNode != NULL ) { postorder(treeNode->getLeft()); postorder(treeNode->getRight()); cout << *(treeNode->getInfo())<<" "; } }

In the postorder, the input argument is a pointer to the TreeNode. If the node is not NULL, we traverse the left tree first. After that we will traverse the right tree and print the node value from where we started.

As all of these above routines are function so we will call them as:

cout << "inorder: "; preorder( root); cout << "inorder: "; inorder( root ); cout << "postorder: "; postorder( root );

Here the root represents the root node of the tree. The size of the tree does not matter as the complete tree will be printed in preorder, inorder and postorder. Let’s discuss an example to see the working of these routines.

Example

Let’s have a look on the following tree.

3 5 7 9 4 17 16 20 18 15 14

This is the same tree we have been using previously. Here we want to traverse the tree. In the bottom of the figure, the numbers are printed with the help of preorder method. These numbers are as 14 4 3 9 7 5 15 18 16 17 20. Now take these numbers and traverse the tree. In the preorder method, we print the root, followed by traversing of the left subtree and the right subtree respectively. As the value of the root node is 14, so it will be printed first of all. After printing the value of the root node, we call the preorder for the left node which is 4. Forget the node 14 as the root is 4 now. So the value 4 is printed and we call the preorder again with the left sub tree i.e. the node with value 3. Now the root is the node with value 3. We will print its value before going to its left. The left side of node with value 3 is NULL. Preorder will be called if condition is false. In this case, no action will be taken. Now the preorder of the left subtree is finished. As the right subtree of this node is also NULL, so there is no need of any action. Now the left subtree of the node with value 4 is complete. The method preorder will be called for the right subtree of the node with value 4. So we call the preorder with the right node of the node with value 4. Here, the root is the node with value 9 that is printed. We will call its left subtree where the node value is 7. It will be followed by its left subtree i.e. node 5 which will be printed.

In the preorder method, we take the root i.e. 14 in this case. Its value is printed, followed by its left subtree and so on. This activity takes us to the extreme left node. Then we back track and print the right subtrees.

Let’s try to understand the inorder method from the following statement.

When we call the inorder function, the numbers will be printed as 3 4 5 7 9 14 15 16 17 18 20. You might have noted that these numbers are in sorted order. When we build the tree, there was no need of sorting the numbers. But here we have the sorted numbers in the inorder method. While inserting a new node, we compare it with the existing ones and place the new node on the left or right side. So during the process of running the inorder on the binary tree, a programmer gets the numbers sorted. Therefore we have a sorting mechanism. If we are given some number, it will not be difficult to get them sorted. This is a new sorting algorithm for you. It is very simple. Just make BST with these numbers and run the inorder traversal. The numbers obtained from the inorder traversal are sorted.

In the inorder method, we do not print the value of node first. We go to traverse its left subtree. When we come back from the left subtree and print the value of this node. Afterwards, we go to the right subtree. Now consider the above tree of figure 13.4. If we start from number 14, it will not be printed. Then we will go to the left subtree. The left subtree is itself a tree. The number 4 is its root. Now being at the node 4, we again look if there is any left subtree of this node. If it has a left subtree, we will not print 4 and call the inorder method on its left subtree. As there is a left subtree of 4 that consists a single node i.e. 3, we go to that node. Now we call the inorder of node 3 to see if there is a subtree of it. As there is no left subtree of 3, the if statement that checks if the node is not NULL will become false. Here the recursive calls will not be executed. We will come back from the call and print the number 3. Thus the first number that is printed in the inorder traversal is 3. After printing 3, we go to its right subtree that is also NULL. So we come back to node 3. Now as we have traversed the left and right subtrees of 3 which itself is a left subtree of 4, thus we have traversed the left subtree of 4. Now we print the value 4 and go to the right subtree of 4. We come at node 9. Before printing 9 we go to its left subtree that leads us to node 7. In this subtree with root 7, we will not print 7. Now we will go to its left subtree which is node 5. We look for the left subtree of 5 which is NULL. Now we print the value 5. Thus, we have so far printed the numbers 3, 4 and 5. Now we come back to 7. After traversing its left subtree, we print the number 7. The right subtree of 7 is NULL. Thus finally, we have traversed the whole tree whose root is 7. It is a left subtree of 9. Now, we will come back to 9 and print it (as its left subtree has been traversed). The right subtree of 9 is NULL. So there is no need to traverse it. Thus the whole tree having 9 as its root has been traversed. So we come back to the root 4, the right subtree of which is the tree with root 9. Resultantly, the left and right subtree of 4 have been traversed now. Thus we have printed the numbers 3, 4, 5, 7 and 9. Now from here (i.e. node 4) we go to the node 14 whose left subtree has been traversed. Now we print 14 and go to its right subtree. The right subtree of 14 has 15 as the root. From here, we repeat the same process earlier carried out in case of the left subtree of 14. As a result of this process, the numbers 15, 16, 17, 18 and 20 are printed. Thus we get the numbers stored in the tree in a sorted order. And we get the numbers printed as 3, 4, 5, 7, 9, 14, 15, 16, 17, 18, 20. This sorted order is only in the inorder traversal.

When we apply the post order traversal on the same tree, the numbers we get are not in sorted order. The numbers got through postorder traversal are in the following order:

3 5 7 9 4 17 16 20 18 15 14

We see that these are not in a sorted order. In order to delete the node from BST, click here.

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