## Saturday, December 26, 2015

### Find all possible binary trees with given Inorder Traversal - GeeksforGeeks

Find all possible binary trees with given Inorder Traversal - GeeksforGeeks
Given an array that represents Inorder Traversal, find all possible Binary Trees with the given Inorder traversal and print their preorder traversals.
Let given inorder traversal be in[]. In the given traversal, all nodes in left subtree of in[i] must appear before it and in right subtree must appear after it. So when we consider in[i] as root, all elements from in[0] to in[i-1] will be in left subtree and in[i+1] to n-1 will be in right subtree. If in[0] to in[i-1] can form xdifferent trees and in[i+1] to in[n-1] can from y different trees then we will have x*y total trees when in[i] as root. So we simply iterate from 0 to n-1 for root. For every node in[i], recursively find different left and right subtrees. If we take a closer look, we can notice that the count is basically n’th Catalan number. We have discussed different approaches to find n’th Catalan number here.
The idea is to maintain a list of roots of all Binary Trees. Recursively construct all possible left and right subtrees. Create a tree for every pair of left and right subtree and add the tree to list. Below is detailed algorithm.
```1) Initialize list of Binary Trees as empty.
2) For every element in[i] where i varies from 0 to n-1,
do following
......a)  Create a new node with key as 'arr[i]',
let this node be 'node'
......b)  Recursively construct list of all left subtrees.
......c)  Recursively construct list of all right subtrees.
3) Iterate for all left subtrees
a) For current leftsubtree, iterate for all right subtrees
'node' to list.```
`class` `Node {`
`    ``int` `data;`
`    ``Node left, right;`
`    ``public` `Node(``int` `item) {`
`        ``data = item;`
`        ``left = ``null``;`
`        ``right = ``null``;`
`    ``}`
`}`
`/* Class to print Level Order Traversal */`
`class` `BinaryTree {`
`    ``Node root;`
`    ``// A utility function to do preorder traversal of BST`
`    ``void` `preOrder(Node node) {`
`        ``if` `(node != ``null``) {`
`            ``System.out.print(node.data + ``" "`    `);`
`            ``preOrder(node.left);`
`            ``preOrder(node.right);`
`        ``}`
`    ``}`
`    ``// Function for constructing all possible trees with`
`    ``// given inorder traversal stored in an array from`
`    ``// arr[start] to arr[end]. This function returns a`
`    ``// vector of trees.`
`    ``Vector<Node> getTrees(``int` `arr[], ``int` `start, ``int` `end) {`
`        ``// List to store all possible trees`
`        ``Vector<Node> trees= ``new` `Vector<Node>();`
`        ``/* if start > end then subtree will be empty so`
`         ``returning NULL in the list */`
`        ``if` `(start > end) {`
`            ``trees.add(``null``);`
`            ``return` `trees;`
`        ``}`
`        ``/* Iterating through all values from start to end`
`         ``for constructing left and right subtree`
`         ``recursively */`
`        ``for` `(``int` `i = start; i <= end; i++) {`
`            ``/* Constructing left subtree */`
`            ``Vector<Node> ltrees = getTrees(arr, start, i - ``1``);`
`            `
`            ``/* Constructing right subtree */`
`            ``Vector<Node> rtrees = getTrees(arr, i + ``1``, end);`
`            ``/* Now looping through all left and right subtrees`
`             ``and connecting them to ith root below */`
`            ``for` `(``int` `j = ``0``; j < ltrees.size(); j++) {`
`                ``for` `(``int` `k = ``0``; k < rtrees.size(); k++) {`
`                    ``// Making arr[i] as root`
`                    ``Node node = ``new` `Node(arr[i]);`
`                    ``// Connecting left subtree`
`                    ``node.left = ltrees.get(j);`
`                    ``// Connecting right subtree`
`                    ``node.right = rtrees.get(k);`
`                    ``// Adding this tree to list`
`                    ``trees.add(node);`
`                ``}`
`            ``}`
`        ``}`
`        ``return` `trees;`
`    ``}`

`class` `Node:`
`    ``# Utility to create a new node`
`    ``def` `__init__(``self` `, item):`
`        ``self``.key ``=` `item`
`        ``self``.left ``=` `None`
`        ``self``.right ``=` `None`
`# A utility function to do preorder traversal of BST`
`def` `preorder(root):`
`    ``if` `root ``is` `not` `None``:`
`        ``print` `root.key,`
`        ``preorder(root.left)`
`        ``preorder(root.right)`
`# Function for constructing all possible trees with`
`# given inorder traversal stored in an array from`
`# arr[start] to arr[end]. This function returns a`
`# vector of trees.`
`def` `getTrees(arr , start , end):`
`    ``# List to store all possible trees`
`    ``trees ``=` `[] `
`    `
`    ``""" if start > end then subtree will be empty so`
`    ``returning NULL in the list """`
`    ``if` `start > end :`
`        ``trees.append(``None``)`
`        ``return` `trees`
`    `
`    ``""" Iterating through all values from start to end`
`        ``for constructing left and right subtree`
`        ``recursively """`
`    ``for` `i ``in` `range``(start , end``+``1``):`
`        ``# Constructing left subtree`
`        ``ltrees ``=` `getTrees(arr , start , i``-``1``)`
`        `
`        ``# Constructing right subtree`
`        ``rtrees ``=` `getTrees(arr , i``+``1` `, end)`
`        `
`        ``""" Looping through all left and right subtrees `
`        ``and connecting to ith root below"""`
`        ``for` `j ``in` `ltrees :`
`            ``for` `k ``in` `rtrees :`
`                ``# Making arr[i]  as root`
`                ``node  ``=` `Node(arr[i])`
`    `
`                ``# Connecting left subtree`
`                ``node.left ``=` `j  `
`                ``# Connecting right subtree`
`                ``node.right ``=` `k `
`                ``# Adding this tree to list`
`                ``trees.append(node)`
`    ``return` `trees`
`# Driver program to test above function`
`inp ``=` `[``4` `, ``5``, ``7``]`
`n ``=` `len``(inp)`
`trees ``=` `getTrees(inp , ``0` `, n``-``1``)`
`print` `"Preorder traversals of different possible\`
` ``Binary Trees are "`
`for` `i ``in` `trees :`
`    ``preorder(i);`
`    ``print` `""`

Related: Construct all possible BSTs for keys 1 to N