# Construct a Binary Search Tree from given postorder

Given postorder traversal of a binary search tree, construct the BST.

For example, if the given traversal is {1, 7, 5, 50, 40, 10}, then following tree should be constructed and root of the tree should be returned.

```     10
/   \
5     40
/  \      \
1    7      50```

## Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Method 1 ( O(n^2) time complexity )
The last element of postorder traversal is always root. We first construct the root. Then we find the index of last element which is smaller than root. Let the index be ‘i’. The values between 0 and ‘i’ are part of left subtree, and the values between ‘i+1’ and ‘n-2’ are part of right subtree. Divide given post[] at index “i” and recur for left and right sub-trees.
For example in {1, 7, 5, 40, 50, 10}, 10 is the last element, so we make it root. Now we look for the last element smaller than 10, we find 5. So we know the structure of BST is as following.

```             10
/    \
/      \
{1, 7, 5}       {50, 40}```

We recursively follow above steps for subarrays {1, 7, 5} and {40, 50}, and get the complete tree.

Method 2 ( O(n) time complexity )
The trick is to set a range {min .. max} for every node. Initialize the range as {INT_MIN .. INT_MAX}. The last node will definitely be in range, so create root node. To construct the left subtree, set the range as {INT_MIN …root->data}. If a values is in the range {INT_MIN .. root->data}, the values is part part of left subtree. To construct the right subtree, set the range as {root->data .. INT_MAX}.

Following code is used to generate the exact Binary Search Tree of a given post order traversal.

## C++

 `/* A O(n) program for construction of  ` `BST from postorder traversal */` `#include ` `using` `namespace` `std; ` ` `  `/* A binary tree node has data,  ` `pointer to left child and a  ` `pointer to right child */` `struct` `node ` `{ ` `    ``int` `data; ` `    ``struct` `node *left, *right; ` `}; ` ` `  `// A utility function to create a node ` `struct` `node* newNode (``int` `data) ` `{ ` `    ``struct` `node* temp = ` `(``struct` `node *) ``malloc``(``sizeof``(``struct` `node)); ` ` `  `    ``temp->data = data; ` `    ``temp->left = temp->right = NULL; ` ` `  `    ``return` `temp; ` `} ` ` `  `// A recursive function to construct  ` `// BST from post[]. postIndex is used  ` `// to keep track of index in post[]. ` `struct` `node* constructTreeUtil(``int` `post[], ``int``* postIndex, ` `                               ``int` `key, ``int` `min, ``int` `max,  ` `                               ``int` `size) ` `{ ` `    ``// Base case ` `    ``if` `(*postIndex < 0) ` `        ``return` `NULL; ` ` `  `    ``struct` `node* root = NULL; ` ` `  `    ``// If current element of post[] is  ` `    ``// in range, then only it is part ` `    ``// of current subtree ` `    ``if` `(key > min && key < max) ` `    ``{ ` `        ``// Allocate memory for root of this  ` `        ``// subtree and decrement *postIndex ` `        ``root = newNode(key); ` `        ``*postIndex = *postIndex - 1; ` ` `  `        ``if` `(*postIndex >= 0) ` `        ``{ ` ` `  `        ``// All nodes which are in range {key..max}  ` `        ``// will go in right subtree, and first such  ` `        ``// node will be root of right subtree. ` `        ``root->right = constructTreeUtil(post, postIndex,  ` `                                        ``post[*postIndex], ` `                                        ``key, max, size ); ` ` `  `        ``// Construct the subtree under root ` `        ``// All nodes which are in range {min .. key}  ` `        ``// will go in left subtree, and first such ` `        ``// node will be root of left subtree. ` `        ``root->left = constructTreeUtil(post, postIndex, ` `                                       ``post[*postIndex], ` `                                       ``min, key, size ); ` `        ``} ` `    ``} ` `    ``return` `root; ` `} ` ` `  `// The main function to construct BST  ` `// from given postorder traversal. ` `// This function mainly uses constructTreeUtil() ` `struct` `node *constructTree (``int` `post[],  ` `                            ``int` `size) ` `{ ` `    ``int` `postIndex = size-1; ` `    ``return` `constructTreeUtil(post, &postIndex,  ` `                             ``post[postIndex], ` `                             ``INT_MIN, INT_MAX, size); ` `} ` ` `  `// A utility function to print ` `// inorder traversal of a Binary Tree ` `void` `printInorder (``struct` `node* node) ` `{ ` `    ``if` `(node == NULL) ` `        ``return``; ` `    ``printInorder(node->left); ` `    ``cout << node->data << ``" "``; ` `    ``printInorder(node->right); ` `} ` ` `  `// Driver Code ` `int` `main () ` `{ ` `    ``int` `post[] = {1, 7, 5, 50, 40, 10}; ` `    ``int` `size = ``sizeof``(post) / ``sizeof``(post); ` ` `  `    ``struct` `node *root = constructTree(post, size); ` ` `  `    ``cout << ``"Inorder traversal of "` `        ``<< ``"the constructed tree: \n"``; ` `    ``printInorder(root); ` ` `  `    ``return` `0; ` `} ` ` `  `// This code is contributed ` `// by Akanksha Rai `

## C

 `/* A O(n) program for construction of BST from ` `   ``postorder traversal */` `#include ` `#include ` `#include ` ` `  `/* A binary tree node has data, pointer to left child ` `   ``and a pointer to right child */` `struct` `node ` `{ ` `    ``int` `data; ` `    ``struct` `node *left, *right; ` `}; ` ` `  `// A utility function to create a node ` `struct` `node* newNode (``int` `data) ` `{ ` `    ``struct` `node* temp = ` `        ``(``struct` `node *) ``malloc``( ``sizeof``(``struct` `node)); ` ` `  `    ``temp->data = data; ` `    ``temp->left = temp->right = NULL; ` ` `  `    ``return` `temp; ` `} ` ` `  `// A recursive function to construct BST from post[].  ` `// postIndex is used to keep track of index in post[]. ` `struct` `node* constructTreeUtil(``int` `post[], ``int``* postIndex, ` `                         ``int` `key, ``int` `min, ``int` `max, ``int` `size) ` `{ ` `    ``// Base case ` `    ``if` `(*postIndex < 0) ` `        ``return` `NULL; ` ` `  `    ``struct` `node* root = NULL; ` ` `  `    ``// If current element of post[] is in range, then ` `    ``// only it is part of current subtree ` `    ``if` `(key > min && key < max) ` `    ``{ ` `        ``// Allocate memory for root of this subtree and decrement ` `        ``// *postIndex ` `        ``root = newNode(key); ` `        ``*postIndex = *postIndex - 1; ` ` `  `        ``if` `(*postIndex >= 0) ` `        ``{ ` ` `  `          ``// All nodes which are in range {key..max} will go in right ` `          ``// subtree, and first such node will be root of right subtree. ` `          ``root->right = constructTreeUtil(post, postIndex, post[*postIndex], ` `                                          ``key, max, size ); ` ` `  `          ``// Construct the subtree under root ` `          ``// All nodes which are in range {min .. key} will go in left ` `          ``// subtree, and first such node will be root of left subtree. ` `          ``root->left = constructTreeUtil(post, postIndex, post[*postIndex], ` `                                         ``min, key, size ); ` `        ``} ` `    ``} ` `    ``return` `root; ` `} ` ` `  `// The main function to construct BST from given postorder ` `// traversal. This function mainly uses constructTreeUtil() ` `struct` `node *constructTree (``int` `post[], ``int` `size) ` `{ ` `    ``int` `postIndex = size-1; ` `    ``return` `constructTreeUtil(post, &postIndex, post[postIndex], ` `                             ``INT_MIN, INT_MAX, size); ` `} ` ` `  `// A utility function to print inorder traversal of a Binary Tree ` `void` `printInorder (``struct` `node* node) ` `{ ` `    ``if` `(node == NULL) ` `        ``return``; ` `    ``printInorder(node->left); ` `    ``printf``(``"%d "``, node->data); ` `    ``printInorder(node->right); ` `} ` ` `  `// Driver program to test above functions ` `int` `main () ` `{ ` `    ``int` `post[] = {1, 7, 5, 50, 40, 10}; ` `    ``int` `size = ``sizeof``(post) / ``sizeof``(post); ` ` `  `    ``struct` `node *root = constructTree(post, size); ` ` `  `    ``printf``(``"Inorder traversal of the constructed tree: \n"``); ` `    ``printInorder(root); ` ` `  `    ``return` `0; ` `} `

## Java

 `/* A O(n) program for construction of BST from ` `   ``postorder traversal */` ` `  ` ``/* A binary tree node has data, pointer to left child ` `   ``and a pointer to right child */` `class` `Node  ` `{ ` `    ``int` `data; ` `    ``Node left, right; ` ` `  `    ``Node(``int` `data)  ` `    ``{ ` `        ``this``.data = data; ` `        ``left = right = ``null``; ` `    ``} ` `} ` ` `  `// Class containing variable that keeps a track of overall ` `// calculated postindex ` `class` `Index  ` `{ ` `    ``int` `postindex = ``0``; ` `} ` ` `  `class` `BinaryTree  ` `{ ` `    ``// A recursive function to construct BST from post[].  ` `    ``// postIndex is used to keep track of index in post[]. ` `    ``Node constructTreeUtil(``int` `post[], Index postIndex, ` `            ``int` `key, ``int` `min, ``int` `max, ``int` `size)  ` `    ``{ ` `        ``// Base case ` `        ``if` `(postIndex.postindex < ``0``) ` `            ``return` `null``; ` ` `  `        ``Node root = ``null``; ` ` `  `        ``// If current element of post[] is in range, then ` `        ``// only it is part of current subtree ` `        ``if` `(key > min && key < max)  ` `        ``{ ` `            ``// Allocate memory for root of this subtree and decrement ` `            ``// *postIndex ` `            ``root = ``new` `Node(key); ` `            ``postIndex.postindex = postIndex.postindex - ``1``; ` ` `  `            ``if` `(postIndex.postindex > ``0``)  ` `            ``{ ` `                ``// All nodes which are in range {key..max} will go in  ` `                ``// right subtree, and first such node will be root of right ` `                ``// subtree ` `                ``root.right = constructTreeUtil(post, postIndex,  ` `                        ``post[postIndex.postindex],key, max, size); ` ` `  `                ``// Construct the subtree under root ` `                ``// All nodes which are in range {min .. key} will go in left ` `                ``// subtree, and first such node will be root of left subtree. ` `                ``root.left = constructTreeUtil(post, postIndex,  ` `                        ``post[postIndex.postindex],min, key, size); ` `            ``} ` `        ``} ` `        ``return` `root; ` `    ``} ` ` `  `    ``// The main function to construct BST from given postorder ` `    ``// traversal. This function mainly uses constructTreeUtil() ` `    ``Node constructTree(``int` `post[], ``int` `size)  ` `    ``{ ` `        ``Index index = ``new` `Index(); ` `        ``index.postindex = size - ``1``; ` `        ``return` `constructTreeUtil(post, index, post[index.postindex], ` `                ``Integer.MIN_VALUE, Integer.MAX_VALUE, size); ` `    ``} ` ` `  `    ``// A utility function to print inorder traversal of a Binary Tree ` `    ``void` `printInorder(Node node)  ` `    ``{ ` `        ``if` `(node == ``null``) ` `            ``return``; ` `        ``printInorder(node.left); ` `        ``System.out.print(node.data + ``" "``); ` `        ``printInorder(node.right); ` `    ``} ` ` `  `    ``// Driver program to test above functions ` `    ``public` `static` `void` `main(String[] args)  ` `    ``{ ` `        ``BinaryTree tree = ``new` `BinaryTree(); ` `        ``int` `post[] = ``new` `int``[]{``1``, ``7``, ``5``, ``50``, ``40``, ``10``}; ` `        ``int` `size = post.length; ` ` `  `        ``Node root = tree.constructTree(post, size); ` ` `  `        ``System.out.println(``"Inorder traversal of the constructed tree:"``); ` `        ``tree.printInorder(root); ` `    ``} ` `} ` ` `  `// This code has been contributed by Mayank Jaiswal  `

## Python3

 `# A O(n) program for construction of BST  ` `# from postorder traversal  ` `INT_MIN ``=` `-``2``*``*``31` `INT_MAX ``=` `2``*``*``31` ` `  `# A binary tree node has data, pointer to  ` `# left child and a pointer to right child ` `# A utility function to create a node  ` `class` `newNode:  ` `    ``def` `__init__(``self``, data):  ` `        ``self``.data ``=` `data  ` `        ``self``.left ``=` `self``.right ``=` `None` `         `  `# A recursive function to construct  ` `# BST from post[]. postIndex is used  ` `# to keep track of index in post[].  ` `def` `constructTreeUtil(post, postIndex,  ` `                      ``key, ``min``, ``max``, size):  ` ` `  `    ``# Base case  ` `    ``if` `(postIndex[``0``] < ``0``): ` `        ``return` `None` ` `  `    ``root ``=` `None` ` `  `    ``# If current element of post[] is  ` `    ``# in range, then only it is part  ` `    ``# of current subtree  ` `    ``if` `(key > ``min` `and` `key < ``max``) : ` `     `  `        ``# Allocate memory for root of this  ` `        ``# subtree and decrement *postIndex  ` `        ``root ``=` `newNode(key)  ` `        ``postIndex[``0``] ``=` `postIndex[``0``] ``-` `1` ` `  `        ``if` `(postIndex[``0``] >``=` `0``) : ` `         `  `            ``# All nodes which are in range key.. ` `            ``# max will go in right subtree, and  ` `            ``# first such node will be root of  ` `            ``# right subtree.  ` `            ``root.right ``=` `constructTreeUtil(post, postIndex, ` `                                           ``post[postIndex[``0``]],  ` `                                           ``key, ``max``, size )  ` ` `  `            ``# Construct the subtree under root  ` `            ``# All nodes which are in range min ..  ` `            ``# key will go in left subtree, and  ` `            ``# first such node will be root of  ` `            ``# left subtree.  ` `            ``root.left ``=` `constructTreeUtil(post, postIndex, ` `                                          ``post[postIndex[``0``]],  ` `                                          ``min``, key, size )  ` `         `  `    ``return` `root  ` ` `  `# The main function to construct BST  ` `# from given postorder traversal. This ` `# function mainly uses constructTreeUtil()  ` `def` `constructTree (post, size) : ` ` `  `    ``postIndex ``=` `[size``-``1``] ` `    ``return` `constructTreeUtil(post, postIndex,  ` `                             ``post[postIndex[``0``]], ` `                             ``INT_MIN, INT_MAX, size)  ` ` `  `# A utility function to prinorder ` `# traversal of a Binary Tree  ` `def` `printInorder (node) : ` ` `  `    ``if` `(node ``=``=` `None``) : ` `        ``return` `    ``printInorder(node.left)  ` `    ``print``(node.data, end ``=` `" "``)  ` `    ``printInorder(node.right)  ` ` `  `# Driver Code  ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``post ``=` `[``1``, ``7``, ``5``, ``50``, ``40``, ``10``] ` `    ``size ``=` `len``(post)  ` ` `  `    ``root ``=` `constructTree(post, size)  ` ` `  `    ``print``(``"Inorder traversal of the"``,  ` `                ``"constructed tree: "``)  ` `    ``printInorder(root) ` ` `  `# This code is contributed  ` `# by SHUBHAMSINGH10 `

## C#

 `using` `System; ` `/* A O(n) program for  ` `construction of BST from  ` `postorder traversal */` ` `  `/* A binary tree node has data, ` `pointer to left child and a ` ` ``pointer to right child */` `class` `Node  ` `{  ` `    ``public` `int` `data;  ` `    ``public` `Node left, right;  ` ` `  `    ``public` `Node(``int` `data)  ` `    ``{  ` `        ``this``.data = data;  ` `        ``left = right = ``null``;  ` `    ``}  ` `}  ` ` `  `// Class containing variable  ` `// that keeps a track of overall  ` `// calculated postindex  ` `class` `Index  ` `{  ` `    ``public` `int` `postindex = 0;  ` `}  ` ` `  `public` `class` `BinaryTree  ` `{  ` `    ``// A recursive function to  ` `    ``// construct BST from post[].  ` `    ``// postIndex is used to  ` `    ``// keep track of index in post[].  ` `    ``Node constructTreeUtil(``int` `[]post, Index postIndex,  ` `                    ``int` `key, ``int` `min, ``int` `max, ``int` `size)  ` `    ``{  ` `        ``// Base case  ` `        ``if` `(postIndex.postindex < 0)  ` `            ``return` `null``;  ` ` `  `        ``Node root = ``null``;  ` ` `  `        ``// If current element of post[] is in range, then  ` `        ``// only it is part of current subtree  ` `        ``if` `(key > min && key < max)  ` `        ``{  ` `            ``// Allocate memory for root of   ` `            ``// this subtree and decrement *postIndex  ` `            ``root = ``new` `Node(key);  ` `            ``postIndex.postindex = postIndex.postindex - 1;  ` ` `  `            ``if` `(postIndex.postindex > 0)  ` `            ``{  ` `                ``// All nodes which are in range ` `                ``// {key..max} will go in right subtree, ` `                ``// and first such node will be root of  ` `                ``// right subtree  ` `                ``root.right = constructTreeUtil(post, postIndex,  ` `                        ``post[postIndex.postindex], key, max, size);  ` ` `  `                ``// Construct the subtree under root  ` `                ``// All nodes which are in range ` `                ``// {min .. key} will go in left  ` `                ``// subtree, and first such node ` `                ``// will be root of left subtree.  ` `                ``root.left = constructTreeUtil(post, postIndex,  ` `                        ``post[postIndex.postindex],min, key, size);  ` `            ``}  ` `        ``}  ` `        ``return` `root;  ` `    ``}  ` ` `  `    ``// The main function to construct ` `    ``// BST from given postorder traversal. ` `    ``// This function mainly uses constructTreeUtil()  ` `    ``Node constructTree(``int` `[]post, ``int` `size)  ` `    ``{  ` `        ``Index index = ``new` `Index();  ` `        ``index.postindex = size - 1;  ` `        ``return` `constructTreeUtil(post, index, ` `                        ``post[index.postindex],  ` `                        ``int``.MinValue, ``int``.MaxValue, size);  ` `    ``}  ` ` `  `    ``// A utility function to print ` `    ``// inorder traversal of a Binary Tree  ` `    ``void` `printInorder(Node node)  ` `    ``{  ` `        ``if` `(node == ``null``)  ` `            ``return``;  ` `        ``printInorder(node.left);  ` `        ``Console.Write(node.data + ``" "``);  ` `        ``printInorder(node.right);  ` `    ``}  ` ` `  `    ``// Driver code  ` `    ``public` `static` `void` `Main(String[] args)  ` `    ``{  ` `        ``BinaryTree tree = ``new` `BinaryTree();  ` `        ``int` `[]post = ``new` `int``[]{1, 7, 5, 50, 40, 10};  ` `        ``int` `size = post.Length;  ` ` `  `        ``Node root = tree.constructTree(post, size);  ` ` `  `        ``Console.WriteLine(``"Inorder traversal of"` `+ ` `                            ``"the constructed tree:"``);  ` `        ``tree.printInorder(root);  ` `    ``}  ` `}  ` ` `  `// This code has been contributed by PrinciRaj1992 `

Output:

```Inorder traversal of the constructed tree:
1 5 7 10 40 50 ```

Note that the output to the program will always be a sorted sequence as we are printing the inorder traversal of a Binary Search Tree.