# Construct a Binary Tree from Postorder and Inorder

Given Postorder and Inorder traversals, construct the tree.
Examples:

```Input:
in[]   = {2, 1, 3}
post[] = {2, 3, 1}

Output: Root of below tree
1
/   \
2     3

Input:
in[]   = {4, 8, 2, 5, 1, 6, 3, 7}
post[] = {8, 4, 5, 2, 6, 7, 3, 1}

Output: Root of below tree
1
/     \
2        3
/    \   /   \
4     5   6    7
\
8
```

We have already discussed construction of tree from Inorder and Preorder traversals. The idea is similar.
Let us see the process of constructing tree from in[] = {4, 8, 2, 5, 1, 6, 3, 7} and post[] = {8, 4, 5, 2, 6, 7, 3, 1}
1) We first find the last node in post[]. The last node is “1”, we know this value is root as root always appear in the end of postorder traversal.
2) We search “1” in in[] to find left and right subtrees of root. Everything on left of “1” in in[] is in left subtree and everything on right is in right subtree.

```         1
/    \
[4, 8, 2, 5]   [6, 3, 7]

```

3) We recur the above process for following two.
….b) Recur for in[] = {6, 3, 7} and post[] = {6, 7, 3}
…….Make the created tree as right child of root.
….a) Recur for in[] = {4, 8, 2, 5} and post[] = {8, 4, 5, 2}.
…….Make the created tree as left child of root.
Below is the implementation of above idea. One important observation is, we recursively call for right subtree before left subtree as we decrease index of postorder index whenever we create a new node.

 `/* C++ program to construct tree using inorder and ` `   ``postorder traversals */` `#include ` ` `  `using` `namespace` `std; ` ` `  `/* A binary tree node has data, pointer to left ` `   ``child and a pointer to right child */` `struct` `Node { ` `    ``int` `data; ` `    ``Node *left, *right; ` `}; ` ` `  `// Utility function to create a new node ` `Node* newNode(``int` `data); ` ` `  `/* Prototypes for utility functions */` `int` `search(``int` `arr[], ``int` `strt, ``int` `end, ``int` `value); ` ` `  `/* Recursive function to construct binary of size n ` `   ``from  Inorder traversal in[] and Postorder traversal ` `   ``post[].  Initial values of inStrt and inEnd should ` `   ``be 0 and n -1.  The function doesn't do any error ` `   ``checking for cases where inorder and postorder ` `   ``do not form a tree */` `Node* buildUtil(``int` `in[], ``int` `post[], ``int` `inStrt, ` `                ``int` `inEnd, ``int``* pIndex) ` `{ ` `    ``// Base case ` `    ``if` `(inStrt > inEnd) ` `        ``return` `NULL; ` ` `  `    ``/* Pick current node from Postorder traversal using ` `       ``postIndex and decrement postIndex */` `    ``Node* node = newNode(post[*pIndex]); ` `    ``(*pIndex)--; ` ` `  `    ``/* If this node has no children then return */` `    ``if` `(inStrt == inEnd) ` `        ``return` `node; ` ` `  `    ``/* Else find the index of this node in Inorder ` `       ``traversal */` `    ``int` `iIndex = search(in, inStrt, inEnd, node->data); ` ` `  `    ``/* Using index in Inorder traversal, construct left and ` `       ``right subtress */` `    ``node->right = buildUtil(in, post, iIndex + 1, inEnd, pIndex); ` `    ``node->left = buildUtil(in, post, inStrt, iIndex - 1, pIndex); ` ` `  `    ``return` `node; ` `} ` ` `  `// This function mainly initializes index of root ` `// and calls buildUtil() ` `Node* buildTree(``int` `in[], ``int` `post[], ``int` `n) ` `{ ` `    ``int` `pIndex = n - 1; ` `    ``return` `buildUtil(in, post, 0, n - 1, &pIndex); ` `} ` ` `  `/* Function to find index of value in arr[start...end] ` `   ``The function assumes that value is postsent in in[] */` `int` `search(``int` `arr[], ``int` `strt, ``int` `end, ``int` `value) ` `{ ` `    ``int` `i; ` `    ``for` `(i = strt; i <= end; i++) { ` `        ``if` `(arr[i] == value) ` `            ``break``; ` `    ``} ` `    ``return` `i; ` `} ` ` `  `/* Helper function that allocates a new node */` `Node* newNode(``int` `data) ` `{ ` `    ``Node* node = (Node*)``malloc``(``sizeof``(Node)); ` `    ``node->data = data; ` `    ``node->left = node->right = NULL; ` `    ``return` `(node); ` `} ` ` `  `/* This funtcion is here just to test  */` `void` `preOrder(Node* node) ` `{ ` `    ``if` `(node == NULL) ` `        ``return``; ` `    ``printf``(``"%d "``, node->data); ` `    ``preOrder(node->left); ` `    ``preOrder(node->right); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `in[] = { 4, 8, 2, 5, 1, 6, 3, 7 }; ` `    ``int` `post[] = { 8, 4, 5, 2, 6, 7, 3, 1 }; ` `    ``int` `n = ``sizeof``(in) / ``sizeof``(in[0]); ` ` `  `    ``Node* root = buildTree(in, post, n); ` ` `  `    ``cout << ``"Preorder of the constructed tree : \n"``; ` `    ``preOrder(root); ` ` `  `    ``return` `0; ` `} `

 `// Java program to construct a tree using inorder ` `// and postorder traversals ` ` `  `/* A binary tree node has data, pointer to left ` `   ``child and a pointer to right child */` `class` `Node { ` `    ``int` `data; ` `    ``Node left, right; ` ` `  `    ``public` `Node(``int` `data) ` `    ``{ ` `        ``this``.data = data; ` `        ``left = right = ``null``; ` `    ``} ` `} ` ` `  `// Class Index created to implement pass by reference of Index ` `class` `Index { ` `    ``int` `index; ` `} ` ` `  `class` `BinaryTree { ` `    ``/* Recursive function to construct binary of size n ` `       ``from  Inorder traversal in[] and Postrder traversal ` `       ``post[].  Initial values of inStrt and inEnd should ` `       ``be 0 and n -1.  The function doesn't do any error ` `       ``checking for cases where inorder and postorder ` `       ``do not form a tree */` `    ``Node buildUtil(``int` `in[], ``int` `post[], ``int` `inStrt, ` `                   ``int` `inEnd, Index pIndex) ` `    ``{ ` `        ``// Base case ` `        ``if` `(inStrt > inEnd) ` `            ``return` `null``; ` ` `  `        ``/* Pick current node from Postrder traversal using ` `           ``postIndex and decrement postIndex */` `        ``Node node = ``new` `Node(post[pIndex.index]); ` `        ``(pIndex.index)--; ` ` `  `        ``/* If this node has no children then return */` `        ``if` `(inStrt == inEnd) ` `            ``return` `node; ` ` `  `        ``/* Else find the index of this node in Inorder ` `           ``traversal */` `        ``int` `iIndex = search(in, inStrt, inEnd, node.data); ` ` `  `        ``/* Using index in Inorder traversal, construct left and ` `           ``right subtress */` `        ``node.right = buildUtil(in, post, iIndex + ``1``, inEnd, pIndex); ` `        ``node.left = buildUtil(in, post, inStrt, iIndex - ``1``, pIndex); ` ` `  `        ``return` `node; ` `    ``} ` ` `  `    ``// This function mainly initializes index of root ` `    ``// and calls buildUtil() ` `    ``Node buildTree(``int` `in[], ``int` `post[], ``int` `n) ` `    ``{ ` `        ``Index pIndex = ``new` `Index(); ` `        ``pIndex.index = n - ``1``; ` `        ``return` `buildUtil(in, post, ``0``, n - ``1``, pIndex); ` `    ``} ` ` `  `    ``/* Function to find index of value in arr[start...end] ` `       ``The function assumes that value is postsent in in[] */` `    ``int` `search(``int` `arr[], ``int` `strt, ``int` `end, ``int` `value) ` `    ``{ ` `        ``int` `i; ` `        ``for` `(i = strt; i <= end; i++) { ` `            ``if` `(arr[i] == value) ` `                ``break``; ` `        ``} ` `        ``return` `i; ` `    ``} ` ` `  `    ``/* This funtcion is here just to test  */` `    ``void` `preOrder(Node node) ` `    ``{ ` `        ``if` `(node == ``null``) ` `            ``return``; ` `        ``System.out.print(node.data + ``" "``); ` `        ``preOrder(node.left); ` `        ``preOrder(node.right); ` `    ``} ` ` `  `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``BinaryTree tree = ``new` `BinaryTree(); ` `        ``int` `in[] = ``new` `int``[] { ``4``, ``8``, ``2``, ``5``, ``1``, ``6``, ``3``, ``7` `}; ` `        ``int` `post[] = ``new` `int``[] { ``8``, ``4``, ``5``, ``2``, ``6``, ``7``, ``3``, ``1` `}; ` `        ``int` `n = in.length; ` `        ``Node root = tree.buildTree(in, post, n); ` `        ``System.out.println(``"Preorder of the constructed tree : "``); ` `        ``tree.preOrder(root); ` `    ``} ` `} ` ` `  `// This code has been contributed by Mayank Jaiswal(mayank_24) `

 `# Python3 program to construct tree using  ` `# inorder and postorder traversals  ` ` `  `# Helper function that allocates  ` `# a new node  ` `class` `newNode: ` `    ``def` `__init__(``self``, data): ` `        ``self``.data ``=` `data  ` `        ``self``.left ``=` `self``.right ``=` `None` ` `  `# Recursive function to construct binary  ` `# of size n from Inorder traversal in[]  ` `# and Postorder traversal post[]. Initial  ` `# values of inStrt and inEnd should be  ` `# 0 and n -1. The function doesn't do any  ` `# error checking for cases where inorder  ` `# and postorder do not form a tree  ` `def` `buildUtil(In, post, inStrt, inEnd, pIndex): ` `     `  `    ``# Base case  ` `    ``if` `(inStrt > inEnd):  ` `        ``return` `None` ` `  `    ``# Pick current node from Postorder traversal  ` `    ``# using postIndex and decrement postIndex  ` `    ``node ``=` `newNode(post[pIndex[``0``]])  ` `    ``pIndex[``0``] ``-``=` `1` ` `  `    ``# If this node has no children  ` `    ``# then return  ` `    ``if` `(inStrt ``=``=` `inEnd):  ` `        ``return` `node  ` ` `  `    ``# Else find the index of this node  ` `    ``# in Inorder traversal  ` `    ``iIndex ``=` `search(In, inStrt, inEnd, node.data)  ` ` `  `    ``# Using index in Inorder traversal,  ` `    ``# construct left and right subtress  ` `    ``node.right ``=` `buildUtil(In, post, iIndex ``+` `1``,  ` `                                  ``inEnd, pIndex)  ` `    ``node.left ``=` `buildUtil(In, post, inStrt,  ` `                        ``iIndex ``-` `1``, pIndex)  ` ` `  `    ``return` `node ` ` `  `# This function mainly initializes index  ` `# of root and calls buildUtil()  ` `def` `buildTree(In, post, n): ` `    ``pIndex ``=` `[n ``-` `1``]  ` `    ``return` `buildUtil(In, post, ``0``, n ``-` `1``, pIndex) ` ` `  `# Function to find index of value in  ` `# arr[start...end]. The function assumes  ` `# that value is postsent in in[]  ` `def` `search(arr, strt, end, value): ` `    ``i ``=` `0` `    ``for` `i ``in` `range``(strt, end ``+` `1``): ` `        ``if` `(arr[i] ``=``=` `value):  ` `            ``break` `    ``return` `i ` ` `  `# This funtcion is here just to test  ` `def` `preOrder(node): ` `    ``if` `node ``=``=` `None``:  ` `        ``return` `    ``print``(node.data,end``=``" "``) ` `    ``preOrder(node.left)  ` `    ``preOrder(node.right) ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``In ``=` `[``4``, ``8``, ``2``, ``5``, ``1``, ``6``, ``3``, ``7``] ` `    ``post ``=` `[``8``, ``4``, ``5``, ``2``, ``6``, ``7``, ``3``, ``1``]  ` `    ``n ``=` `len``(In) ` ` `  `    ``root ``=` `buildTree(In, post, n)  ` ` `  `    ``print``(``"Preorder of the constructed tree :"``)  ` `    ``preOrder(root) ` ` `  `# This code is contributed by PranchalK `

 `// C# program to construct a tree using  ` `// inorder and postorder traversals  ` `using` `System; ` ` `  `/* A binary tree node has data, pointer  ` `to left child and a pointer to right child */` `public` `class` `Node ` `{ ` `    ``public` `int` `data; ` `    ``public` `Node left, right; ` ` `  `    ``public` `Node(``int` `data) ` `    ``{ ` `        ``this``.data = data; ` `        ``left = right = ``null``; ` `    ``} ` `} ` ` `  `// Class Index created to implement  ` `// pass by reference of Index  ` `public` `class` `Index ` `{ ` `    ``public` `int` `index; ` `} ` ` `  `class` `GFG ` `{ ` `/* Recursive function to construct binary  ` `of size n from Inorder traversal in[] and  ` `Postrder traversal post[]. Initial values  ` `of inStrt and inEnd should be 0 and n -1.  ` `The function doesn't do any error checking  ` `for cases where inorder and postorder do  ` `not form a tree */` `public` `virtual` `Node buildUtil(``int``[] @``in``, ``int``[] post,  ` `                              ``int` `inStrt, ``int` `inEnd,  ` `                              ``Index pIndex) ` `{ ` `    ``// Base case  ` `    ``if` `(inStrt > inEnd) ` `    ``{ ` `        ``return` `null``; ` `    ``} ` ` `  `    ``/* Pick current node from Postrder traversal  ` `    ``using postIndex and decrement postIndex */` `    ``Node node = ``new` `Node(post[pIndex.index]); ` `    ``(pIndex.index)--; ` ` `  `    ``/* If this node has no children  ` `    ``then return */` `    ``if` `(inStrt == inEnd) ` `    ``{ ` `        ``return` `node; ` `    ``} ` ` `  `    ``/* Else find the index of this node  ` `    ``in Inorder traversal */` `    ``int` `iIndex = search(@``in``, inStrt, inEnd, node.data); ` ` `  `    ``/* Using index in Inorder traversal,  ` `    ``construct left and right subtress */` `    ``node.right = buildUtil(@``in``, post, iIndex + 1, ` `                                ``inEnd, pIndex); ` `    ``node.left = buildUtil(@``in``, post, inStrt, ` `                               ``iIndex - 1, pIndex); ` ` `  `    ``return` `node; ` `} ` ` `  `// This function mainly initializes  ` `// index of root and calls buildUtil()  ` `public` `virtual` `Node buildTree(``int``[] @``in``, ` `                              ``int``[] post, ``int` `n) ` `{ ` `    ``Index pIndex = ``new` `Index(); ` `    ``pIndex.index = n - 1; ` `    ``return` `buildUtil(@``in``, post, 0, n - 1, pIndex); ` `} ` ` `  `/* Function to find index of value in  ` `arr[start...end]. The function assumes ` `that value is postsent in in[] */` `public` `virtual` `int` `search(``int``[] arr, ``int` `strt,  ` `                          ``int` `end, ``int` `value) ` `{ ` `    ``int` `i; ` `    ``for` `(i = strt; i <= end; i++) ` `    ``{ ` `        ``if` `(arr[i] == value) ` `        ``{ ` `            ``break``; ` `        ``} ` `    ``} ` `    ``return` `i; ` `} ` ` `  `/* This funtcion is here just to test */` `public` `virtual` `void` `preOrder(Node node) ` `{ ` `    ``if` `(node == ``null``) ` `    ``{ ` `        ``return``; ` `    ``} ` `    ``Console.Write(node.data + ``" "``); ` `    ``preOrder(node.left); ` `    ``preOrder(node.right); ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main(``string``[] args) ` `{ ` `    ``GFG tree = ``new` `GFG(); ` `    ``int``[] @``in` `= ``new` `int``[] {4, 8, 2, 5, 1, 6, 3, 7}; ` `    ``int``[] post = ``new` `int``[] {8, 4, 5, 2, 6, 7, 3, 1}; ` `    ``int` `n = @``in``.Length; ` `    ``Node root = tree.buildTree(@``in``, post, n); ` `    ``Console.WriteLine(``"Preorder of the constructed tree : "``); ` `    ``tree.preOrder(root); ` `} ` `} ` ` `  `// This code is contributed by Shrikant13 `

Output
```Preorder of the constructed tree :
1 2 4 8 5 3 6 7 ```

Time Complexity: O(n2)
Optimized approach: We can optimize the above solution using hashing (unordered_map in C++ or HashMap in Java). We store indexes of inorder traversal in a hash table. So that search can be done O(1) time.

 `/* C++ program to construct tree using inorder and  ` `postorder traversals */` `#include ` ` `  `using` `namespace` `std; ` ` `  `/* A binary tree node has data, pointer to left  ` `child and a pointer to right child */` `struct` `Node { ` `    ``int` `data; ` `    ``Node *left, *right; ` `}; ` ` `  `// Utility function to create a new node ` `Node* newNode(``int` `data); ` ` `  `/* Recursive function to construct binary of size n  ` `from Inorder traversal in[] and Postorder traversal  ` `post[]. Initial values of inStrt and inEnd should  ` `be 0 and n -1. The function doesn't do any error  ` `checking for cases where inorder and postorder  ` `do not form a tree */` `Node* buildUtil(``int` `in[], ``int` `post[], ``int` `inStrt, ` `    ``int` `inEnd, ``int``* pIndex, unordered_map<``int``, ``int``>& mp) ` `{ ` `    ``// Base case ` `    ``if` `(inStrt > inEnd) ` `        ``return` `NULL; ` ` `  `    ``/* Pick current node from Postorder traversal   ` `    ``using postIndex and decrement postIndex */` `    ``int` `curr = post[*pIndex]; ` `    ``Node* node = newNode(curr); ` `    ``(*pIndex)--; ` ` `  `    ``/* If this node has no children then return */` `    ``if` `(inStrt == inEnd) ` `        ``return` `node; ` ` `  `    ``/* Else find the index of this node in Inorder  ` `    ``traversal */` `    ``int` `iIndex = mp[curr]; ` ` `  `    ``/* Using index in Inorder traversal, construct  ` `    ``left and right subtress */` `    ``node->right = buildUtil(in, post, iIndex + 1, ` `                            ``inEnd, pIndex, mp); ` `    ``node->left = buildUtil(in, post, inStrt, ` `                           ``iIndex - 1, pIndex, mp); ` ` `  `    ``return` `node; ` `} ` ` `  `// This function mainly creates an unordered_map, then ` `// calls buildTreeUtil() ` `struct` `Node* buildTree(``int` `in[], ``int` `post[], ``int` `len) ` `{ ` `    ``// Store indexes of all items so that we ` `    ``// we can quickly find later ` `    ``unordered_map<``int``, ``int``> mp; ` `    ``for` `(``int` `i = 0; i < len; i++) ` `        ``mp[in[i]] = i; ` ` `  `    ``int` `index = len - 1; ``// Index in postorder ` `    ``return` `buildUtil(in, post, 0, len - 1, ` `                     ``&index, mp); ` `} ` ` `  `/* Helper function that allocates a new node */` `Node* newNode(``int` `data) ` `{ ` `    ``Node* node = (Node*)``malloc``(``sizeof``(Node)); ` `    ``node->data = data; ` `    ``node->left = node->right = NULL; ` `    ``return` `(node); ` `} ` ` `  `/* This funtcion is here just to test */` `void` `preOrder(Node* node) ` `{ ` `    ``if` `(node == NULL) ` `        ``return``; ` `    ``printf``(``"%d "``, node->data); ` `    ``preOrder(node->left); ` `    ``preOrder(node->right); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `in[] = { 4, 8, 2, 5, 1, 6, 3, 7 }; ` `    ``int` `post[] = { 8, 4, 5, 2, 6, 7, 3, 1 }; ` `    ``int` `n = ``sizeof``(in) / ``sizeof``(in[0]); ` ` `  `    ``Node* root = buildTree(in, post, n); ` ` `  `    ``cout << ``"Preorder of the constructed tree : \n"``; ` `    ``preOrder(root); ` ` `  `    ``return` `0; ` `} `

Output
```Preorder of the constructed tree :
1 2 4 8 5 3 6 7 ```

Time Complexity: O(n)

Another approach

Using stack and set without using recursion.

Below is the implementation of the above idea:

 `// CPP program for above approach ` `#include   ` `using` `namespace` `std;  ` ` `  `/* A binary tree node has data, pointer ` `to left   child and a pointer to right  ` `child */` `struct` `Node  ` `{  ` `  ``int` `data;  ` `  ``Node *left, *right;  ` `  ``Node(``int` `x) ` `  ``{ ` `    ``data = x; ` `    ``left = right = NULL; ` `  ``} ` `}; ` ` `  `/*Tree building function*/` `Node *buildTree(``int` `in[], ``int` `post[], ``int` `n)  ` `{ ` ` `  `  ``// Create Stack of type Node* ` `  ``stack st; ` ` `  `  ``// Create Set of type Node* ` `  ``set s; ` ` `  `  ``// Initialise postIndex with n - 1 ` `  ``int` `postIndex = n - 1; ` ` `  `  ``// Initialise root with NULL ` `  ``Node* root = NULL; ` ` `  `  ``for` `(``int` `p = n - 1, i = n - 1; p>=0; )   ` `  ``{  ` ` `  `    ``// Initialise node with NULL ` `    ``Node* node = NULL;  ` `     `  `    ``// Run do-while loop ` `    ``do` `    ``{  ` ` `  `      ``// Initialise node with ` `      ``// new Node(post[p]);  ` `      ``node = ``new` `Node(post[p]);  ` ` `  `      ``// Check is root is  ` `      ``// equal to NULL ` `      ``if` `(root == NULL)  ` `      ``{  ` `        ``root = node;  ` `      ``}  ` `       `  `      ``// If size of set  ` `      ``// is greater than 0 ` `      ``if` `(st.size() > 0)   ` `      ``{  ` `         `  `        ``// If st.top() is present in the ` `        ``// set s ` `        ``if` `(s.find(st.top()) != s.end())  ` `        ``{  ` `          ``s.erase(st.top());  ` `          ``st.top()->left = node;  ` `          ``st.pop();  ` `        ``}  ` `        ``else` `        ``{  ` `          ``st.top()->right = node;  ` `        ``}  ` `      ``} ` `       `  `      ``st.push(node);  ` `       `  `    ``} ``while` `(post[p--] != in[i] && p >=0);  ` ` `  `    ``node = NULL;  ` `     `  `    ``// If the stack is not empty and ` `    ``// st.top()->data is equal to in[i] ` `    ``while` `(st.size() > 0 && i>=0 &&   ` `           ``st.top()->data == in[i])   ` `    ``{  ` `      ``node = st.top();  ` `       `  `      ``// Pop elements from stack ` `      ``st.pop();  ` `      ``i--;  ` `    ``}  ` `     `  `    ``// if node not equal to NULL ` `    ``if` `(node != NULL)   ` `    ``{  ` `      ``s.insert(node);  ` `      ``st.push(node);  ` `    ``}  ` `  ``}  ` `   `  `  ``// Return root ` `  ``return` `root; ` ` `  `} ` `/* for print preOrder Traversal */` `void` `preOrder(Node* node)  ` `{  ` `  ``if` `(node == NULL)  ` `    ``return``;  ` `  ``printf``(``"%d "``, node->data);  ` `  ``preOrder(node->left);  ` `  ``preOrder(node->right);  ` `} ` ` `  `// Driver Code ` `int` `main()  ` `{ ` ` `  `  ``int` `in[] = { 4, 8, 2, 5, 1, 6, 3, 7 };  ` `  ``int` `post[] = { 8, 4, 5, 2, 6, 7, 3, 1 };  ` `  ``int` `n = ``sizeof``(in) / ``sizeof``(in[0]);  ` ` `  `  ``// Function Call ` `  ``Node* root = buildTree(in, post, n);  ` ` `  `  ``cout << "Preorder of the constructed  ` `                                ``tree : \n";  ` ` `  `  ``// Function Call for preOrder ` `  ``preOrder(root);  ` `  ``return` `0; ` `} `

Output
```Preorder of the constructed tree :
1 2 4 8 5 3 6 7 ```
The provided text passage describes a problem of constructing a binary tree from its postorder and inorder traversals. It includes examples, a detailed explanation of the algorithm, and code implementations in C++, Java, Python, and C#. The passage does not contain any explicit mathematical questions or calculations, but it does describe a computational problem and provides a solution.
no_question
Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
system
# Construct a Binary Tree from Postorder and Inorder

Given Postorder and Inorder traversals, construct the tree.
Examples:

```Input:
in[]   = {2, 1, 3}
post[] = {2, 3, 1}

Output: Root of below tree
1
/   \
2     3

Input:
in[]   = {4, 8, 2, 5, 1, 6, 3, 7}
post[] = {8, 4, 5, 2, 6, 7, 3, 1}

Output: Root of below tree
1
/     \
2        3
/    \   /   \
4     5   6    7
\
8
```

We have already discussed construction of tree from Inorder and Preorder traversals. The idea is similar.
Let us see the process of constructing tree from in[] = {4, 8, 2, 5, 1, 6, 3, 7} and post[] = {8, 4, 5, 2, 6, 7, 3, 1}
1) We first find the last node in post[]. The last node is “1”, we know this value is root as root always appear in the end of postorder traversal.
2) We search “1” in in[] to find left and right subtrees of root. Everything on left of “1” in in[] is in left subtree and everything on right is in right subtree.

```         1
/    \
[4, 8, 2, 5]   [6, 3, 7]

```

3) We recur the above process for following two.
….b) Recur for in[] = {6, 3, 7} and post[] = {6, 7, 3}
…….Make the created tree as right child of root.
….a) Recur for in[] = {4, 8, 2, 5} and post[] = {8, 4, 5, 2}.
…….Make the created tree as left child of root.
Below is the implementation of above idea. One important observation is, we recursively call for right subtree before left subtree as we decrease index of postorder index whenever we create a new node.

 `/* C++ program to construct tree using inorder and ` `   ``postorder traversals */` `#include ` ` `  `using` `namespace` `std; ` ` `  `/* A binary tree node has data, pointer to left ` `   ``child and a pointer to right child */` `struct` `Node { ` `    ``int` `data; ` `    ``Node *left, *right; ` `}; ` ` `  `// Utility function to create a new node ` `Node* newNode(``int` `data); ` ` `  `/* Prototypes for utility functions */` `int` `search(``int` `arr[], ``int` `strt, ``int` `end, ``int` `value); ` ` `  `/* Recursive function to construct binary of size n ` `   ``from  Inorder traversal in[] and Postorder traversal ` `   ``post[].  Initial values of inStrt and inEnd should ` `   ``be 0 and n -1.  The function doesn't do any error ` `   ``checking for cases where inorder and postorder ` `   ``do not form a tree */` `Node* buildUtil(``int` `in[], ``int` `post[], ``int` `inStrt, ` `                ``int` `inEnd, ``int``* pIndex) ` `{ ` `    ``// Base case ` `    ``if` `(inStrt > inEnd) ` `        ``return` `NULL; ` ` `  `    ``/* Pick current node from Postorder traversal using ` `       ``postIndex and decrement postIndex */` `    ``Node* node = newNode(post[*pIndex]); ` `    ``(*pIndex)--; ` ` `  `    ``/* If this node has no children then return */` `    ``if` `(inStrt == inEnd) ` `        ``return` `node; ` ` `  `    ``/* Else find the index of this node in Inorder ` `       ``traversal */` `    ``int` `iIndex = search(in, inStrt, inEnd, node->data); ` ` `  `    ``/* Using index in Inorder traversal, construct left and ` `       ``right subtress */` `    ``node->right = buildUtil(in, post, iIndex + 1, inEnd, pIndex); ` `    ``node->left = buildUtil(in, post, inStrt, iIndex - 1, pIndex); ` ` `  `    ``return` `node; ` `} ` ` `  `// This function mainly initializes index of root ` `// and calls buildUtil() ` `Node* buildTree(``int` `in[], ``int` `post[], ``int` `n) ` `{ ` `    ``int` `pIndex = n - 1; ` `    ``return` `buildUtil(in, post, 0, n - 1, &pIndex); ` `} ` ` `  `/* Function to find index of value in arr[start...end] ` `   ``The function assumes that value is postsent in in[] */` `int` `search(``int` `arr[], ``int` `strt, ``int` `end, ``int` `value) ` `{ ` `    ``int` `i; ` `    ``for` `(i = strt; i <= end; i++) { ` `        ``if` `(arr[i] == value) ` `            ``break``; ` `    ``} ` `    ``return` `i; ` `} ` ` `  `/* Helper function that allocates a new node */` `Node* newNode(``int` `data) ` `{ ` `    ``Node* node = (Node*)``malloc``(``sizeof``(Node)); ` `    ``node->data = data; ` `    ``node->left = node->right = NULL; ` `    ``return` `(node); ` `} ` ` `  `/* This funtcion is here just to test  */` `void` `preOrder(Node* node) ` `{ ` `    ``if` `(node == NULL) ` `        ``return``; ` `    ``printf``(``"%d "``, node->data); ` `    ``preOrder(node->left); ` `    ``preOrder(node->right); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `in[] = { 4, 8, 2, 5, 1, 6, 3, 7 }; ` `    ``int` `post[] = { 8, 4, 5, 2, 6, 7, 3, 1 }; ` `    ``int` `n = ``sizeof``(in) / ``sizeof``(in[0]); ` ` `  `    ``Node* root = buildTree(in, post, n); ` ` `  `    ``cout << ``"Preorder of the constructed tree : \n"``; ` `    ``preOrder(root); ` ` `  `    ``return` `0; ` `} `

 `// Java program to construct a tree using inorder ` `// and postorder traversals ` ` `  `/* A binary tree node has data, pointer to left ` `   ``child and a pointer to right child */` `class` `Node { ` `    ``int` `data; ` `    ``Node left, right; ` ` `  `    ``public` `Node(``int` `data) ` `    ``{ ` `        ``this``.data = data; ` `        ``left = right = ``null``; ` `    ``} ` `} ` ` `  `// Class Index created to implement pass by reference of Index ` `class` `Index { ` `    ``int` `index; ` `} ` ` `  `class` `BinaryTree { ` `    ``/* Recursive function to construct binary of size n ` `       ``from  Inorder traversal in[] and Postrder traversal ` `       ``post[].  Initial values of inStrt and inEnd should ` `       ``be 0 and n -1.  The function doesn't do any error ` `       ``checking for cases where inorder and postorder ` `       ``do not form a tree */` `    ``Node buildUtil(``int` `in[], ``int` `post[], ``int` `inStrt, ` `                   ``int` `inEnd, Index pIndex) ` `    ``{ ` `        ``// Base case ` `        ``if` `(inStrt > inEnd) ` `            ``return` `null``; ` ` `  `        ``/* Pick current node from Postrder traversal using ` `           ``postIndex and decrement postIndex */` `        ``Node node = ``new` `Node(post[pIndex.index]); ` `        ``(pIndex.index)--; ` ` `  `        ``/* If this node has no children then return */` `        ``if` `(inStrt == inEnd) ` `            ``return` `node; ` ` `  `        ``/* Else find the index of this node in Inorder ` `           ``traversal */` `        ``int` `iIndex = search(in, inStrt, inEnd, node.data); ` ` `  `        ``/* Using index in Inorder traversal, construct left and ` `           ``right subtress */` `        ``node.right = buildUtil(in, post, iIndex + ``1``, inEnd, pIndex); ` `        ``node.left = buildUtil(in, post, inStrt, iIndex - ``1``, pIndex); ` ` `  `        ``return` `node; ` `    ``} ` ` `  `    ``// This function mainly initializes index of root ` `    ``// and calls buildUtil() ` `    ``Node buildTree(``int` `in[], ``int` `post[], ``int` `n) ` `    ``{ ` `        ``Index pIndex = ``new` `Index(); ` `        ``pIndex.index = n - ``1``; ` `        ``return` `buildUtil(in, post, ``0``, n - ``1``, pIndex); ` `    ``} ` ` `  `    ``/* Function to find index of value in arr[start...end] ` `       ``The function assumes that value is postsent in in[] */` `    ``int` `search(``int` `arr[], ``int` `strt, ``int` `end, ``int` `value) ` `    ``{ ` `        ``int` `i; ` `        ``for` `(i = strt; i <= end; i++) { ` `            ``if` `(arr[i] == value) ` `                ``break``; ` `        ``} ` `        ``return` `i; ` `    ``} ` ` `  `    ``/* This funtcion is here just to test  */` `    ``void` `preOrder(Node node) ` `    ``{ ` `        ``if` `(node == ``null``) ` `            ``return``; ` `        ``System.out.print(node.data + ``" "``); ` `        ``preOrder(node.left); ` `        ``preOrder(node.right); ` `    ``} ` ` `  `    ``public` `static` `void` `main(String[] args) ` `    ``{ ` `        ``BinaryTree tree = ``new` `BinaryTree(); ` `        ``int` `in[] = ``new` `int``[] { ``4``, ``8``, ``2``, ``5``, ``1``, ``6``, ``3``, ``7` `}; ` `        ``int` `post[] = ``new` `int``[] { ``8``, ``4``, ``5``, ``2``, ``6``, ``7``, ``3``, ``1` `}; ` `        ``int` `n = in.length; ` `        ``Node root = tree.buildTree(in, post, n); ` `        ``System.out.println(``"Preorder of the constructed tree : "``); ` `        ``tree.preOrder(root); ` `    ``} ` `} ` ` `  `// This code has been contributed by Mayank Jaiswal(mayank_24) `

 `# Python3 program to construct tree using  ` `# inorder and postorder traversals  ` ` `  `# Helper function that allocates  ` `# a new node  ` `class` `newNode: ` `    ``def` `__init__(``self``, data): ` `        ``self``.data ``=` `data  ` `        ``self``.left ``=` `self``.right ``=` `None` ` `  `# Recursive function to construct binary  ` `# of size n from Inorder traversal in[]  ` `# and Postorder traversal post[]. Initial  ` `# values of inStrt and inEnd should be  ` `# 0 and n -1. The function doesn't do any  ` `# error checking for cases where inorder  ` `# and postorder do not form a tree  ` `def` `buildUtil(In, post, inStrt, inEnd, pIndex): ` `     `  `    ``# Base case  ` `    ``if` `(inStrt > inEnd):  ` `        ``return` `None` ` `  `    ``# Pick current node from Postorder traversal  ` `    ``# using postIndex and decrement postIndex  ` `    ``node ``=` `newNode(post[pIndex[``0``]])  ` `    ``pIndex[``0``] ``-``=` `1` ` `  `    ``# If this node has no children  ` `    ``# then return  ` `    ``if` `(inStrt ``=``=` `inEnd):  ` `        ``return` `node  ` ` `  `    ``# Else find the index of this node  ` `    ``# in Inorder traversal  ` `    ``iIndex ``=` `search(In, inStrt, inEnd, node.data)  ` ` `  `    ``# Using index in Inorder traversal,  ` `    ``# construct left and right subtress  ` `    ``node.right ``=` `buildUtil(In, post, iIndex ``+` `1``,  ` `                                  ``inEnd, pIndex)  ` `    ``node.left ``=` `buildUtil(In, post, inStrt,  ` `                        ``iIndex ``-` `1``, pIndex)  ` ` `  `    ``return` `node ` ` `  `# This function mainly initializes index  ` `# of root and calls buildUtil()  ` `def` `buildTree(In, post, n): ` `    ``pIndex ``=` `[n ``-` `1``]  ` `    ``return` `buildUtil(In, post, ``0``, n ``-` `1``, pIndex) ` ` `  `# Function to find index of value in  ` `# arr[start...end]. The function assumes  ` `# that value is postsent in in[]  ` `def` `search(arr, strt, end, value): ` `    ``i ``=` `0` `    ``for` `i ``in` `range``(strt, end ``+` `1``): ` `        ``if` `(arr[i] ``=``=` `value):  ` `            ``break` `    ``return` `i ` ` `  `# This funtcion is here just to test  ` `def` `preOrder(node): ` `    ``if` `node ``=``=` `None``:  ` `        ``return` `    ``print``(node.data,end``=``" "``) ` `    ``preOrder(node.left)  ` `    ``preOrder(node.right) ` ` `  `# Driver code  ` `if` `__name__ ``=``=` `'__main__'``: ` `    ``In ``=` `[``4``, ``8``, ``2``, ``5``, ``1``, ``6``, ``3``, ``7``] ` `    ``post ``=` `[``8``, ``4``, ``5``, ``2``, ``6``, ``7``, ``3``, ``1``]  ` `    ``n ``=` `len``(In) ` ` `  `    ``root ``=` `buildTree(In, post, n)  ` ` `  `    ``print``(``"Preorder of the constructed tree :"``)  ` `    ``preOrder(root) ` ` `  `# This code is contributed by PranchalK `

 `// C# program to construct a tree using  ` `// inorder and postorder traversals  ` `using` `System; ` ` `  `/* A binary tree node has data, pointer  ` `to left child and a pointer to right child */` `public` `class` `Node ` `{ ` `    ``public` `int` `data; ` `    ``public` `Node left, right; ` ` `  `    ``public` `Node(``int` `data) ` `    ``{ ` `        ``this``.data = data; ` `        ``left = right = ``null``; ` `    ``} ` `} ` ` `  `// Class Index created to implement  ` `// pass by reference of Index  ` `public` `class` `Index ` `{ ` `    ``public` `int` `index; ` `} ` ` `  `class` `GFG ` `{ ` `/* Recursive function to construct binary  ` `of size n from Inorder traversal in[] and  ` `Postrder traversal post[]. Initial values  ` `of inStrt and inEnd should be 0 and n -1.  ` `The function doesn't do any error checking  ` `for cases where inorder and postorder do  ` `not form a tree */` `public` `virtual` `Node buildUtil(``int``[] @``in``, ``int``[] post,  ` `                              ``int` `inStrt, ``int` `inEnd,  ` `                              ``Index pIndex) ` `{ ` `    ``// Base case  ` `    ``if` `(inStrt > inEnd) ` `    ``{ ` `        ``return` `null``; ` `    ``} ` ` `  `    ``/* Pick current node from Postrder traversal  ` `    ``using postIndex and decrement postIndex */` `    ``Node node = ``new` `Node(post[pIndex.index]); ` `    ``(pIndex.index)--; ` ` `  `    ``/* If this node has no children  ` `    ``then return */` `    ``if` `(inStrt == inEnd) ` `    ``{ ` `        ``return` `node; ` `    ``} ` ` `  `    ``/* Else find the index of this node  ` `    ``in Inorder traversal */` `    ``int` `iIndex = search(@``in``, inStrt, inEnd, node.data); ` ` `  `    ``/* Using index in Inorder traversal,  ` `    ``construct left and right subtress */` `    ``node.right = buildUtil(@``in``, post, iIndex + 1, ` `                                ``inEnd, pIndex); ` `    ``node.left = buildUtil(@``in``, post, inStrt, ` `                               ``iIndex - 1, pIndex); ` ` `  `    ``return` `node; ` `} ` ` `  `// This function mainly initializes  ` `// index of root and calls buildUtil()  ` `public` `virtual` `Node buildTree(``int``[] @``in``, ` `                              ``int``[] post, ``int` `n) ` `{ ` `    ``Index pIndex = ``new` `Index(); ` `    ``pIndex.index = n - 1; ` `    ``return` `buildUtil(@``in``, post, 0, n - 1, pIndex); ` `} ` ` `  `/* Function to find index of value in  ` `arr[start...end]. The function assumes ` `that value is postsent in in[] */` `public` `virtual` `int` `search(``int``[] arr, ``int` `strt,  ` `                          ``int` `end, ``int` `value) ` `{ ` `    ``int` `i; ` `    ``for` `(i = strt; i <= end; i++) ` `    ``{ ` `        ``if` `(arr[i] == value) ` `        ``{ ` `            ``break``; ` `        ``} ` `    ``} ` `    ``return` `i; ` `} ` ` `  `/* This funtcion is here just to test */` `public` `virtual` `void` `preOrder(Node node) ` `{ ` `    ``if` `(node == ``null``) ` `    ``{ ` `        ``return``; ` `    ``} ` `    ``Console.Write(node.data + ``" "``); ` `    ``preOrder(node.left); ` `    ``preOrder(node.right); ` `} ` ` `  `// Driver Code ` `public` `static` `void` `Main(``string``[] args) ` `{ ` `    ``GFG tree = ``new` `GFG(); ` `    ``int``[] @``in` `= ``new` `int``[] {4, 8, 2, 5, 1, 6, 3, 7}; ` `    ``int``[] post = ``new` `int``[] {8, 4, 5, 2, 6, 7, 3, 1}; ` `    ``int` `n = @``in``.Length; ` `    ``Node root = tree.buildTree(@``in``, post, n); ` `    ``Console.WriteLine(``"Preorder of the constructed tree : "``); ` `    ``tree.preOrder(root); ` `} ` `} ` ` `  `// This code is contributed by Shrikant13 `

Output
```Preorder of the constructed tree :
1 2 4 8 5 3 6 7 ```

Time Complexity: O(n2)
Optimized approach: We can optimize the above solution using hashing (unordered_map in C++ or HashMap in Java). We store indexes of inorder traversal in a hash table. So that search can be done O(1) time.

 `/* C++ program to construct tree using inorder and  ` `postorder traversals */` `#include ` ` `  `using` `namespace` `std; ` ` `  `/* A binary tree node has data, pointer to left  ` `child and a pointer to right child */` `struct` `Node { ` `    ``int` `data; ` `    ``Node *left, *right; ` `}; ` ` `  `// Utility function to create a new node ` `Node* newNode(``int` `data); ` ` `  `/* Recursive function to construct binary of size n  ` `from Inorder traversal in[] and Postorder traversal  ` `post[]. Initial values of inStrt and inEnd should  ` `be 0 and n -1. The function doesn't do any error  ` `checking for cases where inorder and postorder  ` `do not form a tree */` `Node* buildUtil(``int` `in[], ``int` `post[], ``int` `inStrt, ` `    ``int` `inEnd, ``int``* pIndex, unordered_map<``int``, ``int``>& mp) ` `{ ` `    ``// Base case ` `    ``if` `(inStrt > inEnd) ` `        ``return` `NULL; ` ` `  `    ``/* Pick current node from Postorder traversal   ` `    ``using postIndex and decrement postIndex */` `    ``int` `curr = post[*pIndex]; ` `    ``Node* node = newNode(curr); ` `    ``(*pIndex)--; ` ` `  `    ``/* If this node has no children then return */` `    ``if` `(inStrt == inEnd) ` `        ``return` `node; ` ` `  `    ``/* Else find the index of this node in Inorder  ` `    ``traversal */` `    ``int` `iIndex = mp[curr]; ` ` `  `    ``/* Using index in Inorder traversal, construct  ` `    ``left and right subtress */` `    ``node->right = buildUtil(in, post, iIndex + 1, ` `                            ``inEnd, pIndex, mp); ` `    ``node->left = buildUtil(in, post, inStrt, ` `                           ``iIndex - 1, pIndex, mp); ` ` `  `    ``return` `node; ` `} ` ` `  `// This function mainly creates an unordered_map, then ` `// calls buildTreeUtil() ` `struct` `Node* buildTree(``int` `in[], ``int` `post[], ``int` `len) ` `{ ` `    ``// Store indexes of all items so that we ` `    ``// we can quickly find later ` `    ``unordered_map<``int``, ``int``> mp; ` `    ``for` `(``int` `i = 0; i < len; i++) ` `        ``mp[in[i]] = i; ` ` `  `    ``int` `index = len - 1; ``// Index in postorder ` `    ``return` `buildUtil(in, post, 0, len - 1, ` `                     ``&index, mp); ` `} ` ` `  `/* Helper function that allocates a new node */` `Node* newNode(``int` `data) ` `{ ` `    ``Node* node = (Node*)``malloc``(``sizeof``(Node)); ` `    ``node->data = data; ` `    ``node->left = node->right = NULL; ` `    ``return` `(node); ` `} ` ` `  `/* This funtcion is here just to test */` `void` `preOrder(Node* node) ` `{ ` `    ``if` `(node == NULL) ` `        ``return``; ` `    ``printf``(``"%d "``, node->data); ` `    ``preOrder(node->left); ` `    ``preOrder(node->right); ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``int` `in[] = { 4, 8, 2, 5, 1, 6, 3, 7 }; ` `    ``int` `post[] = { 8, 4, 5, 2, 6, 7, 3, 1 }; ` `    ``int` `n = ``sizeof``(in) / ``sizeof``(in[0]); ` ` `  `    ``Node* root = buildTree(in, post, n); ` ` `  `    ``cout << ``"Preorder of the constructed tree : \n"``; ` `    ``preOrder(root); ` ` `  `    ``return` `0; ` `} `

Output
```Preorder of the constructed tree :
1 2 4 8 5 3 6 7 ```

Time Complexity: O(n)

Another approach

Using stack and set without using recursion.

Below is the implementation of the above idea:

 `// CPP program for above approach ` `#include   ` `using` `namespace` `std;  ` ` `  `/* A binary tree node has data, pointer ` `to left   child and a pointer to right  ` `child */` `struct` `Node  ` `{  ` `  ``int` `data;  ` `  ``Node *left, *right;  ` `  ``Node(``int` `x) ` `  ``{ ` `    ``data = x; ` `    ``left = right = NULL; ` `  ``} ` `}; ` ` `  `/*Tree building function*/` `Node *buildTree(``int` `in[], ``int` `post[], ``int` `n)  ` `{ ` ` `  `  ``// Create Stack of type Node* ` `  ``stack st; ` ` `  `  ``// Create Set of type Node* ` `  ``set s; ` ` `  `  ``// Initialise postIndex with n - 1 ` `  ``int` `postIndex = n - 1; ` ` `  `  ``// Initialise root with NULL ` `  ``Node* root = NULL; ` ` `  `  ``for` `(``int` `p = n - 1, i = n - 1; p>=0; )   ` `  ``{  ` ` `  `    ``// Initialise node with NULL ` `    ``Node* node = NULL;  ` `     `  `    ``// Run do-while loop ` `    ``do` `    ``{  ` ` `  `      ``// Initialise node with ` `      ``// new Node(post[p]);  ` `      ``node = ``new` `Node(post[p]);  ` ` `  `      ``// Check is root is  ` `      ``// equal to NULL ` `      ``if` `(root == NULL)  ` `      ``{  ` `        ``root = node;  ` `      ``}  ` `       `  `      ``// If size of set  ` `      ``// is greater than 0 ` `      ``if` `(st.size() > 0)   ` `      ``{  ` `         `  `        ``// If st.top() is present in the ` `        ``// set s ` `        ``if` `(s.find(st.top()) != s.end())  ` `        ``{  ` `          ``s.erase(st.top());  ` `          ``st.top()->left = node;  ` `          ``st.pop();  ` `        ``}  ` `        ``else` `        ``{  ` `          ``st.top()->right = node;  ` `        ``}  ` `      ``} ` `       `  `      ``st.push(node);  ` `       `  `    ``} ``while` `(post[p--] != in[i] && p >=0);  ` ` `  `    ``node = NULL;  ` `     `  `    ``// If the stack is not empty and ` `    ``// st.top()->data is equal to in[i] ` `    ``while` `(st.size() > 0 && i>=0 &&   ` `           ``st.top()->data == in[i])   ` `    ``{  ` `      ``node = st.top();  ` `       `  `      ``// Pop elements from stack ` `      ``st.pop();  ` `      ``i--;  ` `    ``}  ` `     `  `    ``// if node not equal to NULL ` `    ``if` `(node != NULL)   ` `    ``{  ` `      ``s.insert(node);  ` `      ``st.push(node);  ` `    ``}  ` `  ``}  ` `   `  `  ``// Return root ` `  ``return` `root; ` ` `  `} ` `/* for print preOrder Traversal */` `void` `preOrder(Node* node)  ` `{  ` `  ``if` `(node == NULL)  ` `    ``return``;  ` `  ``printf``(``"%d "``, node->data);  ` `  ``preOrder(node->left);  ` `  ``preOrder(node->right);  ` `} ` ` `  `// Driver Code ` `int` `main()  ` `{ ` ` `  `  ``int` `in[] = { 4, 8, 2, 5, 1, 6, 3, 7 };  ` `  ``int` `post[] = { 8, 4, 5, 2, 6, 7, 3, 1 };  ` `  ``int` `n = ``sizeof``(in) / ``sizeof``(in[0]);  ` ` `  `  ``// Function Call ` `  ``Node* root = buildTree(in, post, n);  ` ` `  `  ``cout << "Preorder of the constructed  ` `                                ``tree : \n";  ` ` `  `  ``// Function Call for preOrder ` `  ``preOrder(root);  ` `  ``return` `0; ` `} `

Output
```Preorder of the constructed tree :
1 2 4 8 5 3 6 7 ```
user
The provided text passage describes a problem of constructing a binary tree from its postorder and inorder traversals. It includes examples, a detailed explanation of the algorithm, and code implementations in C++, Java, Python, and C#. The passage does not contain any explicit mathematical questions or calculations, but it does describe a computational problem and provides a solution.

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