# Count of strictly increasing and decreasing paths in directed Binary Tree

• Last Updated : 16 Jun, 2022

Given a directed binary tree of N nodes whose edges are directed from the parent to the child, the task is to count the number of strictly increasing and decreasing paths.

Note: A path starts at the root and ends at any leaf.

Examples:

Input: N = 6
tree =             6
/         \
4             7
\          /    \
5      6       8
Output: 10 8
Explanation: For the above binary tree, the strictly increasing paths are :
All the paths consisting of a single node are = 6.
The paths containing two nodes are: 4->5, 7->8, 6->7 = 3.
The paths containing three nodes are: 6->7->8 = 1.
Hence, the count of strictly increasing paths is 6 + 3 + 1 = 10.
For the above binary tree, the strictly decreasing paths are:
All the paths consisting of a single node are = 6.
The paths containing two nodes are: 6->4, 7->6 = 2.
Hence, the count of strictly decreasing paths is 6 + 2 = 8.

Input: N = 5
tree                 15
/
8
\
2
/    \
13     9
Output: 7 8
Explanation:
For the above binary tree, the strictly increasing paths are:
All the paths consisting of a single node are = 5.
The paths containing two nodes are: 2->9, 2->13 = 2.
Hence, the count of strictly increasing paths is 5 + 2 = 7.
For the above binary tree, the strictly decreasing paths are:
All the paths consisting of a single node are = 5.
The paths containing two nodes are:15->8, 8->2 = 2.
The paths containing three nodes are: 15->8->2 = 1.
Hence, the count of strictly decreasing paths is 5 + 2 + 1 = 8.

Approach: The problem can be solved based on the following observation:

If a path starting from any node u is strictly increasing and its value is greater than the value of its parent (say v), then all the strictly increasing paths starting from u are also strictly increasing paths when the starting node is v. The same is true in case of strictly decreasing paths.

From the above observation, the problem can be solved with the help of recursion. Follow the steps below to solve the problem:

• Use a recursive function to traverse the tree starting from the root and for each node return the count of strictly increasing and strictly decreasing paths from that node.
• In each recursion:
• Check if the path from the current node to a child (left or right) is increasing or decreasing.
• Call the recursive function for the child.
• Add the count of increasing or decreasing paths to the total count if the path from the current node to the child is increasing or decreasing respectively.
• Return the final count of increasing or decreasing path obtained for root.

Below is the implementation of the above approach :

## C++

 `// C++ code to implement the approach`   `#include ` `using` `namespace` `std;`   `// Following is the TreeNode` `// class structure:` `template` `<``typename` `T>` `class` `TreeNode {` `public``:` `    ``T data;` `    ``TreeNode* left;` `    ``TreeNode* right;`   `    ``TreeNode(T data)` `    ``{` `        ``this``->data = data;` `        ``left = NULL;` `        ``right = NULL;` `    ``}` `};`   `// Helper function to get the answer` `pair<``int``, ``int``> countPathsHelper(TreeNode<``int``>* root,` `                                ``int``& increase,` `                                ``int``& decrease)` `{` `    ``// If the root is NULL,` `    ``// then no strictly increasing or` `    ``// decreasing path.` `    ``if` `(root == NULL) {` `        ``return` `{ 0, 0 };` `    ``}`   `    ``// Call the function for both` `    ``// the left and right child.` `    ``pair<``int``, ``int``> p1` `        ``= countPathsHelper(root->left,` `                           ``increase, decrease);` `    ``pair<``int``, ``int``> p2` `        ``= countPathsHelper(root->right,` `                           ``increase, decrease);`   `    ``// Initialize 'inc' and 'dec' to 1` `    ``int` `inc = 1, dec = 1;`   `    ``// If the left child is not NULL.` `    ``if` `(root->left != NULL) {`   `        ``// Check if the value is` `        ``// increasing from parent to child.` `        ``if` `(root->data < root->left->data) {`   `            ``// Add the count of strictly` `            ``// increasing paths of` `            ``// child to parent.` `            ``inc += p1.first;` `        ``}`   `        ``// Check if the value is decreasing` `        ``// from parent to child.` `        ``if` `(root->data > root->left->data) {`   `            ``// Add the count of strictly` `            ``// decreasing paths of` `            ``// child to parent.` `            ``dec += p1.second;` `        ``}` `    ``}` `    ``if` `(root->right != NULL) {`   `        ``// Check if the value is` `        ``// increasing from parent to child.` `        ``if` `(root->data < root->right->data) {`   `            ``// Add the count of strictly` `            ``// increasing paths of` `            ``// child to parent.` `            ``inc += p2.first;` `        ``}`   `        ``// Check if the value is` `        ``// decreasing from parent to child` `        ``if` `(root->data > root->right->data) {`   `            ``// Add the count of strictly` `            ``// decreasing paths of` `            ``// child to parent.` `            ``dec += p2.second;` `        ``}` `    ``}`   `    ``// Add the total count of` `    ``// strictly increasing paths to` `    ``// the global strictly increasing` `    ``// paths counter.` `    ``increase += inc;`   `    ``// Add the total count of` `    ``// strictly decreasing paths to` `    ``// the global strictly` `    ``// decreasing paths counter.` `    ``decrease += dec;` `    ``return` `{ inc, dec };` `}`   `// Function to count the paths` `pair<``int``, ``int``> countPaths(TreeNode<``int``>* root)` `{` `    ``// 'increase' stores the` `    ``// total strictly increasing paths.` `    ``int` `increase = 0;`   `    ``// 'decrease' stores the` `    ``// total strictly decreasing paths.` `    ``int` `decrease = 0;`   `    ``countPathsHelper(root, increase, decrease);` `    ``return` `{ increase, decrease };` `}`   `// Driver code` `int` `main()` `{` `    ``int` `N = 6;` `    ``TreeNode<``int``>* root` `        ``= ``new` `TreeNode<``int``>(N);` `    ``root->left = ``new` `TreeNode<``int``>(4);` `    ``root->right = ``new` `TreeNode<``int``>(7);` `    ``root->left->right = ``new` `TreeNode<``int``>(5);` `    ``root->right->left = ``new` `TreeNode<``int``>(6);` `    ``root->right->right = ``new` `TreeNode<``int``>(8);`   `    ``// Function call` `    ``pair<``int``, ``int``> ans = countPaths(root);` `    ``cout << ans.first << ``" "` `<< ans.second;` `    ``return` `0;` `}`

## Java

 `// Java code for the above approach:` `import` `java.util.*; `   `public` `class` `Main {` `  ``static` `class` `Node {` `    ``int` `data;` `    ``Node left;` `    ``Node right;`   `    ``Node(``int` `data_value)` `    ``{` `      ``data = data_value;` `      ``this``.left = ``null``;` `      ``this``.right = ``null``;` `    ``}` `  ``}` `  ``// 'increase' stores the` `  ``// total strictly increasing paths.` `  ``static` `int` `increase;`   `  ``// 'decrease' stores the` `  ``// total strictly decreasing paths.` `  ``static` `int` `decrease;` `  ``// Helper function to get the answer` `  ``static` `int``[] countPathsHelper(Node root)` `  ``{` `    ``// If the root is NULL,` `    ``// then no strictly increasing or` `    ``// decreasing path.` `    ``if` `(root == ``null``) {` `      ``int``[] zero = {``0``,``0``};` `      ``return` `zero;` `    ``}`   `    ``// Call the function for both` `    ``// the left and right child.` `    ``int``[] p1` `      ``= countPathsHelper(root.left);` `    ``int``[] p2` `      ``= countPathsHelper(root.right);`   `    ``// Initialize 'inc' and 'dec' to 1` `    ``int` `inc = ``1``, dec = ``1``;`   `    ``// If the left child is not NULL.` `    ``if` `(root.left != ``null``) {`   `      ``// Check if the value is` `      ``// increasing from parent to child.` `      ``if` `(root.data < root.left.data) {`   `        ``// Add the count of strictly` `        ``// increasing paths of` `        ``// child to parent.` `        ``inc += p1[``0``];` `      ``}`   `      ``// Check if the value is decreasing` `      ``// from parent to child.` `      ``if` `(root.data > root.left.data) {`   `        ``// Add the count of strictly` `        ``// decreasing paths of` `        ``// child to parent.` `        ``dec += p1[``1``];` `      ``}` `    ``}` `    ``if` `(root.right != ``null``) {`   `      ``// Check if the value is` `      ``// increasing from parent to child.` `      ``if` `(root.data < root.right.data) {`   `        ``// Add the count of strictly` `        ``// increasing paths of` `        ``// child to parent.` `        ``inc += p2[``0``];` `      ``}`   `      ``// Check if the value is` `      ``// decreasing from parent to child` `      ``if` `(root.data > root.right.data) {`   `        ``// Add the count of strictly` `        ``// decreasing paths of` `        ``// child to parent.` `        ``dec += p2[``1``];` `      ``}` `    ``}`   `    ``// Add the total count of` `    ``// strictly increasing paths to` `    ``// the global strictly increasing` `    ``// paths counter.` `    ``increase += inc;`   `    ``// Add the total count of` `    ``// strictly decreasing paths to` `    ``// the global strictly` `    ``// decreasing paths counter.` `    ``decrease += dec;` `    ``int``[] ans = {inc,dec};` `    ``return` `ans;` `  ``}`   `  ``// Function to count the paths` `  ``static` `int``[] countPaths(Node root)` `  ``{` `    ``increase=``0``;` `    ``decrease=``0``;` `    ``countPathsHelper(root);` `    ``int``[] ans = {increase,decrease};`   `    ``return` `ans;` `  ``}`   `  ``public` `static` `void` `main(String args[]) {` `    ``int` `N = ``6``;` `    ``Node root = ``new` `Node(N);` `    ``root.left = ``new` `Node(``4``);` `    ``root.right = ``new` `Node(``7``);` `    ``root.left.right = ``new` `Node(``5``);` `    ``root.right.left = ``new` `Node(``6``);` `    ``root.right.right = ``new` `Node(``8``);`   `    ``// Function call` `    ``int``[] ans = countPaths(root);` `    ``System.out.println(ans[``0``] + ``" "` `+ ans[``1``]);` `  ``}` `}`   `// This code has been contributed by Sachin Sahara (sachin801)`

## Python3

 `# Python code for the above approach`   `## structure for a node` `class` `Node:` `    ``def` `__init__(``self``, d):` `        ``self``.data ``=` `d` `        ``self``.left ``=` `None` `        ``self``.right ``=` `None`   `## 'increase' stores the` `## total strictly increasing paths.` `increase ``=` `0`   `## 'decrease' stores the` `## total strictly decreasing paths.` `decrease ``=` `0`   `## Helper function to get the answer` `def` `countPathsHelper(root):`   `    ``global` `increase` `    ``global` `decrease` `    `  `    ``## If the root is NULL,` `    ``## then no strictly increasing or` `    ``## decreasing path.` `    ``if` `(root ``=``=` `None``):` `        ``zero ``=` `[``0``, ``0``]` `        ``return` `zero`   `    ``## Call the function for both` `    ``## the left and right child.` `    ``p1 ``=` `countPathsHelper(root.left)` `    ``p2 ``=` `countPathsHelper(root.right)`   `    ``## Initialize 'inc' and 'dec' to 1` `    ``inc ``=` `1` `    ``dec ``=` `1`   `    ``## If the left child is not NULL.` `    ``if` `(root.left !``=` `None``):`   `        ``## Check if the value is` `        ``## increasing from parent to child.` `        ``if` `(root.data < root.left.data):`   `            ``## Add the count of strictly` `            ``## increasing paths of` `            ``## child to parent.` `            ``inc ``+``=` `p1[``0``];`   `        ``## Check if the value is decreasing` `        ``## from parent to child.` `        ``if` `(root.data > root.left.data):`   `            ``## Add the count of strictly` `            ``## decreasing paths of` `            ``## child to parent.` `            ``dec ``+``=` `p1[``1``]`   `    ``if` `(root.right !``=` `None``):`   `        ``## Check if the value is` `        ``## increasing from parent to child.` `        ``if` `(root.data < root.right.data):`   `            ``## Add the count of strictly` `            ``## increasing paths of` `            ``## child to parent.` `            ``inc ``+``=` `p2[``0``]`   `            ``## Check if the value is` `            ``## decreasing from parent to child` `            ``if` `(root.data > root.right.data):`   `                ``## Add the count of strictly` `                ``## decreasing paths of` `                ``## child to parent.` `                ``dec ``+``=` `p2[``1``]`   `    ``## Add the total count of` `    ``## strictly increasing paths to` `    ``## the global strictly increasing` `    ``## paths counter.` `    ``increase ``+``=` `inc`   `    ``## Add the total count of` `    ``## strictly decreasing paths to` `    ``## the global strictly` `    ``## decreasing paths counter.` `    ``decrease ``+``=` `dec` `    ``ans ``=` `[inc, dec]` `    ``return` `ans`   `## Function to count the paths` `def` `countPaths(root):` `    ``global` `increase` `    ``global` `decrease`   `    ``increase ``=` `0` `    ``decrease ``=` `0`   `    ``countPathsHelper(root)` `    ``ans ``=` `[increase, decrease]`   `    ``return` `ans`   `# Driver Code` `if` `__name__``=``=``'__main__'``:`   `    ``N ``=` `6` `    ``root ``=` `Node(N)` `    ``root.left ``=` `Node(``4``)` `    ``root.right ``=` `Node(``7``)` `    ``root.left.right ``=` `Node(``5``)` `    ``root.right.left ``=` `Node(``6``)` `    ``root.right.right ``=` `Node(``8``)`   `    ``## Function call` `    ``ans ``=` `countPaths(root)` `    ``print``(ans[``0``], ans[``1``])`   `    ``# This code is contributed by entertain2022.`

## C#

 `// C# program to count the number of ` `// strictly increasing and decreasing paths` `using` `System;`   `public` `class` `GFG{` `    `  `    ``public` `class` `Node {` `        ``public` `int` `data;` `        ``public` `Node left;` `        ``public` `Node right;` `     `  `        ``public` `Node(``int` `data_value)` `        ``{` `          ``data = data_value;` `          ``this``.left = ``null``;` `          ``this``.right = ``null``;` `        ``}` `    ``}` `    `  `      ``// 'increase' stores the` `      ``// total strictly increasing paths.` `      ``static` `int` `increase;` `     `  `      ``// 'decrease' stores the` `      ``// total strictly decreasing paths.` `      ``static` `int` `decrease;` `      `  `    ``// Helper function to get the answer` `  ``static` `int``[] countPathsHelper(Node root)` `  ``{` `        ``// If the root is NULL,` `        ``// then no strictly increasing or` `        ``// decreasing path.` `        ``if` `(root == ``null``) {` `          ``int``[] zero = {0,0};` `          ``return` `zero;` `        ``}` `     `  `        ``// Call the function for both` `        ``// the left and right child.` `        ``int``[] p1 = countPathsHelper(root.left);` `        ``int``[] p2 = countPathsHelper(root.right);` `     `  `        ``// Initialize 'inc' and 'dec' to 1` `        ``int` `inc = 1, dec = 1;` `     `  `        ``// If the left child is not NULL.` `        ``if` `(root.left != ``null``) {` `     `  `          ``// Check if the value is` `          ``// increasing from parent to child.` `          ``if` `(root.data < root.left.data) {` `     `  `            ``// Add the count of strictly` `            ``// increasing paths of` `            ``// child to parent.` `            ``inc += p1;` `          ``}` `     `  `          ``// Check if the value is decreasing` `          ``// from parent to child.` `          ``if` `(root.data > root.left.data) {` `     `  `            ``// Add the count of strictly` `            ``// decreasing paths of` `            ``// child to parent.` `            ``dec += p1;` `          ``}` `        ``}` `        ``if` `(root.right != ``null``) {` `     `  `          ``// Check if the value is` `          ``// increasing from parent to child.` `          ``if` `(root.data < root.right.data) {` `     `  `            ``// Add the count of strictly` `            ``// increasing paths of` `            ``// child to parent.` `            ``inc += p2;` `          ``}` `     `  `          ``// Check if the value is` `          ``// decreasing from parent to child` `          ``if` `(root.data > root.right.data) {` `     `  `            ``// Add the count of strictly` `            ``// decreasing paths of` `            ``// child to parent.` `            ``dec += p2;` `          ``}` `        ``}` `     `  `        ``// Add the total count of` `        ``// strictly increasing paths to` `        ``// the global strictly increasing` `        ``// paths counter.` `        ``increase += inc;` `     `  `        ``// Add the total count of` `        ``// strictly decreasing paths to` `        ``// the global strictly` `        ``// decreasing paths counter.` `        ``decrease += dec;` `        ``int``[] ans = {inc,dec};` `        ``return` `ans;` `    ``}` `    `  `      ``// Function to count the paths` `      ``static` `int``[] countPaths(Node root)` `      ``{` `        ``increase=0;` `        ``decrease=0;` `        ``countPathsHelper(root);` `        ``int``[] ans = {increase,decrease};` `     `  `        ``return` `ans;` `      ``}` `    `  `    ``static` `public` `void` `Main (){` `        ``//Code` `        ``int` `N = 6;` `        ``Node root = ``new` `Node(N);` `        ``root.left = ``new` `Node(4);` `        ``root.right = ``new` `Node(7);` `        ``root.left.right = ``new` `Node(5);` `        ``root.right.left = ``new` `Node(6);` `        ``root.right.right = ``new` `Node(8);` `     `  `        ``// Function call` `        ``int``[] ans = countPaths(root);` `        ``Console.Write(ans + ``" "` `+ ans);` `    ``}` `}`   `// This code is contributed by shruti456rawal`

Output

`10 8`

Time Complexity: O(N)
Auxiliary Space: O(N)

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