# Sum of lengths of all paths possible in a given tree

Given a tree with N nodes, the task is to find the sum of lengths of all the paths. Path length for two nodes in the tree is the number of edges on the path and for two adjacent nodes in the tree, the length of the path is 1.

Examples:

```Input:
0
/   \
1      2
/  \
3    4
Output: 18
Paths of length 1 = (0, 1), (0, 2), (1, 3), (1, 4) = 4
Paths of length 2 = (0, 3), (0, 4), (1, 2), (3, 4) = 4
Paths of length 3 = (3, 2), (4, 2) = 2
The sum of lengths of all paths =
(4 * 1) + (4 * 2) + (2 * 3) = 18

Input:
0
/
1
/
2
Output: 4
```

## Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Naive approach: Check all possible paths and then add them to compute the final result. The complexity of this approach will be O(n2).

Efficient approach: It can be noted that each edge in a tree is a bridge. Hence that edge is going to be present in every path possible between the two subtrees that the edge connects.
For example, the edge (1 – 0) is present in every path possible between {1, 3, 4} and {0, 2}, (1 – 0) is used for 6 times that is size of the subtree {1, 3, 4} multiplied by the size of the subtree {0, 2}. So for each edge, compute how many times that edge is going to be considered for the paths going over it. DFS can be used to store the size of the subtree and the contribution of all edges can be computed with another dfs. The complexity of this approach will be O(n).

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` `#include ` `using` `namespace` `std; ` ` `  `const` `int` `sz = 1e5; ` ` `  `// Number of vertices ` `int` `n; ` ` `  `// Adjacency list representation ` `// of the tree ` `vector<``int``> tree[sz]; ` ` `  `// Array that stores the subtree size ` `int` `subtree_size[sz]; ` ` `  `// Array to mark all the ` `// vertices which are visited ` `int` `vis[sz]; ` ` `  `// Utility function to create an ` `// edge between two vertices ` `void` `addEdge(``int` `a, ``int` `b) ` `{ ` ` `  `    ``// Add a to b's list ` `    ``tree[a].push_back(b); ` ` `  `    ``// Add b to a's list ` `    ``tree[b].push_back(a); ` `} ` ` `  `// Function to calculate the subtree size ` `int` `dfs(``int` `node) ` `{ ` ` `  `    ``// Mark visited ` `    ``vis[node] = 1; ` `    ``subtree_size[node] = 1; ` ` `  `    ``// For every adjacent node ` `    ``for` `(``auto` `child : tree[node]) { ` ` `  `        ``// If not already visited ` `        ``if` `(!vis[child]) { ` ` `  `            ``// Recursive call for the child ` `            ``subtree_size[node] += dfs(child); ` `        ``} ` `    ``} ` `    ``return` `subtree_size[node]; ` `} ` ` `  `// Function to calculate the ` `// contribution of each edge ` `void` `contribution(``int` `node, ``int``& ans) ` `{ ` ` `  `    ``// Mark current node as visited ` `    ``vis[node] = ``true``; ` ` `  `    ``// For every adjacent node ` `    ``for` `(``int` `child : tree[node]) { ` ` `  `        ``// If it is not already visisted ` `        ``if` `(!vis[child]) { ` `            ``ans += (subtree_size[child] ` `                    ``* (n - subtree_size[child])); ` `            ``contribution(child, ans); ` `        ``} ` `    ``} ` `} ` ` `  `// Function to return the required sum ` `int` `getSum() ` `{ ` ` `  `    ``// First pass of the dfs ` `    ``memset``(vis, 0, ``sizeof``(vis)); ` `    ``dfs(0); ` ` `  `    ``// Second pass ` `    ``int` `ans = 0; ` `    ``memset``(vis, 0, ``sizeof``(vis)); ` `    ``contribution(0, ans); ` ` `  `    ``return` `ans; ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``n = 5; ` ` `  `    ``addEdge(0, 1); ` `    ``addEdge(0, 2); ` `    ``addEdge(1, 3); ` `    ``addEdge(1, 4); ` ` `  `    ``cout << getSum(); ` ` `  `    ``return` `0; ` `} `

## Python3

 `# Python3 implementation of the approach ` `sz ``=` `10``*``*``5` ` `  `# Number of vertices ` `n ``=` `5` `an ``=` `0` ` `  `# Adjacency list representation ` `# of the tree ` `tree ``=` `[[] ``for` `i ``in` `range``(sz)] ` ` `  `# Array that stores the subtree size ` `subtree_size ``=` `[``0``] ``*` `sz ` ` `  `# Array to mark all the ` `# vertices which are visited ` `vis ``=` `[``0``] ``*` `sz ` ` `  `# Utility function to create an ` `# edge between two vertices ` `def` `addEdge(a, b): ` ` `  `    ``# Add a to b's list ` `    ``tree[a].append(b) ` ` `  `    ``# Add b to a's list ` `    ``tree[b].append(a) ` ` `  `# Function to calculate the subtree size ` `def` `dfs(node): ` `    ``leaf ``=` `True` ` `  `    ``# Mark visited ` `    ``vis[node] ``=` `1` `     `  `    ``# For every adjacent node ` `    ``for` `child ``in` `tree[node]: ` ` `  `        ``# If not already visited ` `        ``if` `(vis[child] ``=``=` `0``): ` `            ``leaf ``=` `False` `            ``dfs(child) ` `             `  `            ``# Recursive call for the child ` `            ``subtree_size[node] ``+``=` `subtree_size[child] ` ` `  `    ``if` `leaf: ` `        ``subtree_size[node] ``=` `1` ` `  `# Function to calculate the ` `# contribution of each edge ` `def` `contribution(node,ans): ` `    ``global` `an ` ` `  `    ``# Mark current node as visited ` `    ``vis[node] ``=` `1` ` `  `    ``# For every adjacent node ` `    ``for` `child ``in` `tree[node]: ` ` `  `        ``# If it is not already visisted ` `        ``if` `(vis[child] ``=``=` `0``): ` `            ``an ``+``=` `(subtree_size[child] ``*`  `              ``(n ``-` `subtree_size[child])) ` `            ``contribution(child, ans) ` ` `  `# Function to return the required sum ` `def` `getSum(): ` ` `  `    ``# First pass of the dfs ` `    ``for` `i ``in` `range``(sz): ` `        ``vis[i] ``=` `0` `    ``dfs(``0``) ` ` `  `    ``# Second pass ` `    ``ans ``=` `0` `    ``for` `i ``in` `range``(sz): ` `        ``vis[i] ``=` `0` ` `  `    ``contribution(``0``, ans) ` ` `  `    ``return` `an ` ` `  `# Driver code ` `n ``=` `5` ` `  `addEdge(``0``, ``1``) ` `addEdge(``0``, ``2``) ` `addEdge(``1``, ``3``) ` `addEdge(``1``, ``4``) ` ` `  `print``(getSum()) ` ` `  `# This code is contributed by Mohit Kumar `

Output:

```18
```

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