# Find the root of the sub-tree whose weighted sum is minimum

Given a tree, and the weights of all the nodes, the task is to find the root of the sub-tree whose weighted sum is minimum.

**Examples:**

Input:

Output:5

Weight of sub-tree for parent 1 = ((-1) + (5) + (-2) + (-1) + (3)) = 4

Weight of sub-tree for parent 2 = ((5) + (-1) + (3)) = 7

Weight of sub-tree for parent 3 = -1

Weight of sub-tree for parent 4 = 3

Weight of sub-tree for parent 5 = -2

Node 5 gives the minimum sub-tree weighted sum.

**Approach:** Perform dfs on the tree, and for every node calculate the sub-tree weighted sum rooted at the current node then find the minimum sum value for a node.

Below is the implementation of the above approach:

## C++

`// C++ implementation of the approach` `#include <bits/stdc++.h>` `using` `namespace` `std;` `int` `ans = 0, mini = INT_MAX;` `vector<` `int` `> graph[100];` `vector<` `int` `> weight(100);` `// Function to perform dfs and update the tree` `// such that every node's weight is the sum of` `// the weights of all the nodes in the sub-tree` `// of the current node including itself` `void` `dfs(` `int` `node, ` `int` `parent)` `{` ` ` `for` `(` `int` `to : graph[node]) {` ` ` `if` `(to == parent)` ` ` `continue` `;` ` ` `dfs(to, node);` ` ` `// Calculating the weighted` ` ` `// sum of the subtree` ` ` `weight[node] += weight[to];` ` ` `}` `}` `// Function to find the node` `// having minimum sub-tree sum` `void` `findMin(` `int` `n)` `{` ` ` `// For every node` ` ` `for` `(` `int` `i = 1; i <= n; i++) {` ` ` `// If current node's weight` ` ` `// is minimum so far` ` ` `if` `(mini > weight[i]) {` ` ` `mini = weight[i];` ` ` `ans = i;` ` ` `}` ` ` `}` `}` `// Driver code` `int` `main()` `{` ` ` `int` `n = 5;` ` ` `// Weights of the node` ` ` `weight[1] = -1;` ` ` `weight[2] = 5;` ` ` `weight[3] = -1;` ` ` `weight[4] = 3;` ` ` `weight[5] = -2;` ` ` `// Edges of the tree` ` ` `graph[1].push_back(2);` ` ` `graph[2].push_back(3);` ` ` `graph[2].push_back(4);` ` ` `graph[1].push_back(5);` ` ` `dfs(1, 1);` ` ` `findMin(n);` ` ` `cout << ans;` ` ` `return` `0;` `}` |

## Java

`// Java implementation of the approach` `import` `java.util.*;` `class` `GFG` `{` ` ` `static` `int` `ans = ` `0` `, mini = Integer.MAX_VALUE;` ` ` ` ` `@SuppressWarnings` `(` `"unchecked"` `)` ` ` `static` `Vector<Integer>[] graph = ` `new` `Vector[` `100` `];` ` ` `static` `Integer[] weight = ` `new` `Integer[` `100` `];` ` ` `// Function to perform dfs and update the tree` ` ` `// such that every node's weight is the sum of` ` ` `// the weights of all the nodes in the sub-tree` ` ` `// of the current node including itself` ` ` `static` `void` `dfs(` `int` `node, ` `int` `parent)` ` ` `{` ` ` `for` `(` `int` `to : graph[node])` ` ` `{` ` ` `if` `(to == parent)` ` ` `continue` `;` ` ` `dfs(to, node);` ` ` `// Calculating the weighted` ` ` `// sum of the subtree` ` ` `weight[node] += weight[to];` ` ` `}` ` ` `}` ` ` `// Function to find the node` ` ` `// having minimum sub-tree sum x` ` ` `static` `void` `findMin(` `int` `n)` ` ` `{` ` ` `// For every node` ` ` `for` `(` `int` `i = ` `1` `; i <= n; i++)` ` ` `{` ` ` `// If current node's weight x` ` ` `// is minimum so far` ` ` `if` `(mini > weight[i])` ` ` `{` ` ` `mini = weight[i];` ` ` `ans = i;` ` ` `}` ` ` `}` ` ` `}` ` ` `// Driver code` ` ` `public` `static` `void` `main(String[] args)` ` ` `{` ` ` ` ` `int` `n = ` `5` `;` ` ` `for` `(` `int` `i = ` `0` `; i < ` `100` `; i++)` ` ` `graph[i] = ` `new` `Vector<Integer>();` ` ` ` ` `// Weights of the node` ` ` `weight[` `1` `] = -` `1` `;` ` ` `weight[` `2` `] = ` `5` `;` ` ` `weight[` `3` `] = -` `1` `;` ` ` `weight[` `4` `] = ` `3` `;` ` ` `weight[` `5` `] = -` `2` `;` ` ` `// Edges of the tree` ` ` `graph[` `1` `].add(` `2` `);` ` ` `graph[` `2` `].add(` `3` `);` ` ` `graph[` `2` `].add(` `4` `);` ` ` `graph[` `1` `].add(` `5` `);` ` ` `dfs(` `1` `, ` `1` `);` ` ` `findMin(n);` ` ` `System.out.print(ans);` ` ` `}` `}` `// This code is contributed by shubhamsingh10` |

## C#

`// C# implementation of the approach` `using` `System;` `using` `System.Collections.Generic;` `class` `GFG` `{` ` ` `static` `int` `ans = 0, mini = ` `int` `.MaxValue;` ` ` `static` `List<` `int` `>[] graph = ` `new` `List<` `int` `>[100];` ` ` `static` `int` `[] weight = ` `new` `int` `[100];` ` ` ` ` `// Function to perform dfs and update the tree` ` ` `// such that every node's weight is the sum of` ` ` `// the weights of all the nodes in the sub-tree` ` ` `// of the current node including itself` ` ` `static` `void` `dfs(` `int` `node, ` `int` `parent)` ` ` `{` ` ` `foreach` `(` `int` `to ` `in` `graph[node])` ` ` `{` ` ` `if` `(to == parent)` ` ` `continue` `;` ` ` `dfs(to, node);` ` ` ` ` `// Calculating the weighted` ` ` `// sum of the subtree` ` ` `weight[node] += weight[to];` ` ` `}` ` ` `}` ` ` ` ` `// Function to find the node` ` ` `// having minimum sub-tree sum x` ` ` `static` `void` `findMin(` `int` `n)` ` ` `{` ` ` ` ` `// For every node` ` ` `for` `(` `int` `i = 1; i <= n; i++)` ` ` `{` ` ` ` ` `// If current node's weight x` ` ` `// is minimum so far` ` ` `if` `(mini > weight[i])` ` ` `{` ` ` `mini = weight[i];` ` ` `ans = i;` ` ` `}` ` ` `}` ` ` `}` ` ` ` ` `// Driver code` ` ` `public` `static` `void` `Main(String[] args)` ` ` `{` ` ` ` ` `int` `n = 5;` ` ` `for` `(` `int` `i = 0; i < 100; i++)` ` ` `graph[i] = ` `new` `List<` `int` `>();` ` ` ` ` `// Weights of the node` ` ` `weight[1] = -1;` ` ` `weight[2] = 5;` ` ` `weight[3] = -1;` ` ` `weight[4] = 3;` ` ` `weight[5] = -2;` ` ` ` ` `// Edges of the tree` ` ` `graph[1].Add(2);` ` ` `graph[2].Add(3);` ` ` `graph[2].Add(4);` ` ` `graph[1].Add(5);` ` ` ` ` `dfs(1, 1);` ` ` `findMin(n);` ` ` ` ` `Console.Write(ans);` ` ` `}` `}` `// This code is contributed by Rajput-Ji` |

## Python3

`# Python3 implementation of the approach` `ans ` `=` `0` `mini ` `=` `2` `*` `*` `32` `graph ` `=` `[[] ` `for` `i ` `in` `range` `(` `100` `)]` `weight ` `=` `[` `0` `]` `*` `100` `# Function to perform dfs and update the tree` `# such that every node's weight is the sum of` `# the weights of all the nodes in the sub-tree` `# of the current node including itself` `def` `dfs(node, parent):` ` ` `global` `mini, graph, weight, ans` ` ` `for` `to ` `in` `graph[node]:` ` ` `if` `(to ` `=` `=` `parent):` ` ` `continue` ` ` `dfs(to, node)` ` ` ` ` `# Calculating the weighted` ` ` `# sum of the subtree` ` ` `weight[node] ` `+` `=` `weight[to]` ` ` `# Function to find the node` `# having minimum sub-tree sum` `def` `findMin(n):` ` ` `global` `mini, graph, weight, ans` ` ` ` ` `# For every node` ` ` `for` `i ` `in` `range` `(` `1` `, n ` `+` `1` `):` ` ` ` ` `# If current node's weight` ` ` `# is minimum so far` ` ` `if` `(mini > weight[i]):` ` ` `mini ` `=` `weight[i]` ` ` `ans ` `=` `i` `# Driver code` `n ` `=` `5` `# Weights of the node` `weight[` `1` `] ` `=` `-` `1` `weight[` `2` `] ` `=` `5` `weight[` `3` `] ` `=` `-` `1` `weight[` `4` `] ` `=` `3` `weight[` `5` `] ` `=` `-` `2` `# Edges of the tree` `graph[` `1` `].append(` `2` `)` `graph[` `2` `].append(` `3` `)` `graph[` `2` `].append(` `4` `)` `graph[` `1` `].append(` `5` `)` `dfs(` `1` `, ` `1` `)` `findMin(n)` `print` `(ans)` `# This code is contributed by SHUBHAMSINGH10` |

## Javascript

`<script>` ` ` `// Javascript implementation of the approach` ` ` `let ans = 0;` `let mini = Number.MAX_VALUE;` `let graph = ` `new` `Array(100);` `let weight = ` `new` `Array(100);` `for` `(let i = 0; i < 100; i++)` `{` ` ` `graph[i] = [];` ` ` `weight[i] = 0;` `}` `// Function to perform dfs and update the tree` `// such that every node's weight is the sum of` `// the weights of all the nodes in the sub-tree` `// of the current node including itself` `function` `dfs(node, parent)` `{` ` ` `for` `(let to = 0; to < graph[node].length; to++)` ` ` `{` ` ` `if` `(graph[node][to] == parent)` ` ` `continue` ` ` ` ` `dfs(graph[node][to], node); ` ` ` ` ` `// Calculating the weighted` ` ` `// sum of the subtree` ` ` `weight[node] += weight[graph[node][to]];` ` ` `}` `}` `// Function to find the node` `// having minimum sub-tree sum` `function` `findMin(n)` `{` ` ` `// For every node` ` ` `for` `(let i = 1; i <= n; i++)` ` ` `{` ` ` ` ` `// If current node's weight` ` ` `// is minimum so far` ` ` `if` `(mini > weight[i])` ` ` `{` ` ` `mini = weight[i];` ` ` `ans = i;` ` ` `}` ` ` `}` `}` `// Driver code` `let n = 5;` `// Weights of the node` `weight[1] = -1;` `weight[2] = 5;` `weight[3] = -1;` `weight[4] = 3;` `weight[5] = -2;` `// Edges of the tree` `graph[1].push(2);` `graph[2].push(3);` `graph[2].push(4);` `graph[1].push(5);` `dfs(1, 1);` `findMin(n);` `document.write(ans);` `// This code is contributed by Dharanendra L V.` ` ` `</script>` |

**Output:**

5

__Complexity Analysis:__

**Time Complexity :**O(N).

In dfs, every node of the tree is processed once and hence the complexity due to the dfs is O(N) if there are total N nodes in the tree. Therefore, the time complexity is O(N).**Auxiliary Space :**O(n).

Recursion stack.