# Find triplet such that number of nodes connecting these triplets is maximum

Given a Tree with N nodes, the task is to find a triplet of nodes (a, b, c) such that the number of nodes covered in the path connecting these nodes is maximum. (Count a node only once).

Examples:

Input: N = 4
Edge Set:
1 2
1 3
1 4
Output: (2, 3, 4)
(2, 3, 4) as path between (2, 3) covers nodes 2, 1, 3 and path between (3, 4) covers nodes 3, 1, 4. Hence all nodes are covered. The Red Path in Tree denotes the path between 2 to 3 node which covers node 1, 2, 3. The green path denotes path between (3, 4) which covers node 3, 1, 4.

Input: N = 9
Edge Set :
1 2
2 3
3 4
4 5
5 6
2 7
7 8
4 9
Output: (6, 8, 1)

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

Approach:

• One important point to notice is, two of the points in triplet must be the end of diameter of tree to cover maximum of the points.
• We need to find the longest length branch stick to the diameter.
• Now for 3rd node, apply DFS along with maintaining the depth for each node (DFS in all directions other than on the Diameter Path Selected ) to all the nodes present on the path of Diameter, the node which is at the farthest distance, would be considered as the 3rd node, as it covers the maximum node other than already covered by the Diameter. Diameter of Tree using DFS

Below is the implementation of the above approach:

## C++

 `// C++ implementation of the approach ` `#include ` `#define ll long long int ` `#define MAX 100005 ` `using` `namespace` `std; ` `vector<``int``> adjacent[MAX]; ` `bool` `visited[MAX]; ` ` `  `// To store the required nodes ` `int` `startnode, endnode, thirdnode; ` ` `  `int` `maxi = -1, N; ` ` `  `// Parent array to retrace the nodes ` `int` `parent[MAX]; ` ` `  `// Visited array to prevent DFS ` `// in direction on Diameter path ` `bool` `vis[MAX]; ` ` `  `// DFS function to find the startnode ` `void` `dfs(``int` `u, ``int` `count) ` `{ ` `    ``visited[u] = ``true``; ` `    ``int` `temp = 0; ` `    ``for` `(``int` `i = 0; i < adjacent[u].size(); i++) { ` `        ``if` `(!visited[adjacent[u][i]]) { ` `            ``temp++; ` `            ``dfs(adjacent[u][i], count + 1); ` `        ``} ` `    ``} ` ` `  `    ``if` `(temp == 0) { ` `        ``if` `(maxi < count) { ` `            ``maxi = count; ` `            ``startnode = u; ` `        ``} ` `    ``} ` `} ` ` `  `// DFS function to find the endnode ` `// of diameter and maintain the parent array ` `void` `dfs1(``int` `u, ``int` `count) ` `{ ` `    ``visited[u] = ``true``; ` `    ``int` `temp = 0; ` `    ``for` `(``int` `i = 0; i < adjacent[u].size(); i++) { ` `        ``if` `(!visited[adjacent[u][i]]) { ` `            ``temp++; ` `            ``parent[adjacent[u][i]] = u; ` `            ``dfs1(adjacent[u][i], count + 1); ` `        ``} ` `    ``} ` ` `  `    ``if` `(temp == 0) { ` `        ``if` `(maxi < count) { ` `            ``maxi = count; ` `            ``endnode = u; ` `        ``} ` `    ``} ` `} ` ` `  `// DFS function to find the end node ` `// of the Longest Branch to Diameter ` `void` `dfs2(``int` `u, ``int` `count) ` `{ ` `    ``visited[u] = ``true``; ` `    ``int` `temp = 0; ` `    ``for` `(``int` `i = 0; i < adjacent[u].size(); i++) { ` `        ``if` `(!visited[adjacent[u][i]] ` `            ``&& !vis[adjacent[u][i]]) { ` `            ``temp++; ` `            ``dfs2(adjacent[u][i], count + 1); ` `        ``} ` `    ``} ` `    ``if` `(temp == 0) { ` `        ``if` `(maxi < count) { ` `            ``maxi = count; ` `            ``thirdnode = u; ` `        ``} ` `    ``} ` `} ` ` `  `// Function to find the required nodes ` `void` `findNodes() ` `{ ` `    ``// To find start node of diameter ` `    ``dfs(1, 0); ` ` `  `    ``for` `(``int` `i = 0; i <= N; i++) ` `        ``visited[i] = ``false``; ` ` `  `    ``maxi = -1; ` ` `  `    ``// To find end node of diameter ` `    ``dfs1(startnode, 0); ` ` `  `    ``for` `(``int` `i = 0; i <= N; i++) ` `        ``visited[i] = ``false``; ` ` `  `    ``// x is the end node of diameter ` `    ``int` `x = endnode; ` `    ``vis[startnode] = ``true``; ` ` `  `    ``// Mark all the nodes on diameter ` `    ``// using back tracking ` `    ``while` `(x != startnode) { ` `        ``vis[x] = ``true``; ` `        ``x = parent[x]; ` `    ``} ` ` `  `    ``maxi = -1; ` ` `  `    ``// Find the end node of longest ` `    ``// branch to diameter ` `    ``for` `(``int` `i = 1; i <= N; i++) { ` `        ``if` `(vis[i]) ` `            ``dfs2(i, 0); ` `    ``} ` `} ` ` `  `// Driver code ` `int` `main() ` `{ ` `    ``N = 4; ` `    ``adjacent.push_back(2); ` `    ``adjacent.push_back(1); ` `    ``adjacent.push_back(3); ` `    ``adjacent.push_back(1); ` `    ``adjacent.push_back(4); ` `    ``adjacent.push_back(1); ` ` `  `    ``findNodes(); ` ` `  `    ``cout << ``"("` `<< startnode << ``", "` `<< endnode ` `         ``<< ``", "` `<< thirdnode << ``")"``; ` ` `  `    ``return` `0; ` `} `

## Java

 `// Java implementation of the approach ` `import` `java.util.*; ` ` `  `class` `GFG ` `{ ` `static` `int` `MAX = ``100005``; ` `static` `Vector> adjacent = ``new`  `       ``Vector> (); ` `static` `boolean` `visited[] = ``new` `boolean``[MAX]; ` ` `  `// To store the required nodes ` `static` `int` `startnode, endnode, thirdnode; ` ` `  `static` `int` `maxi = -``1``, N; ` ` `  `// Parent array to retrace the nodes ` `static` `int` `parent[] = ``new` `int``[MAX]; ` ` `  `// Visited array to prevent DFS ` `// in direction on Diameter path ` `static` `boolean` `vis[] = ``new` `boolean``[MAX]; ` ` `  `// DFS function to find the startnode ` `static` `void` `dfs(``int` `u, ``int` `count) ` `{ ` `    ``visited[u] = ``true``; ` `    ``int` `temp = ``0``; ` `    ``for` `(``int` `i = ``0``; i < adjacent.get(u).size(); i++) ` `    ``{ ` `        ``if` `(!visited[adjacent.get(u).get(i)])  ` `        ``{ ` `            ``temp++; ` `            ``dfs(adjacent.get(u).get(i), count + ``1``); ` `        ``} ` `    ``} ` ` `  `    ``if` `(temp == ``0``)  ` `    ``{ ` `        ``if` `(maxi < count) ` `        ``{ ` `            ``maxi = count; ` `            ``startnode = u; ` `        ``} ` `    ``} ` `} ` ` `  `// DFS function to find the endnode ` `// of diameter and maintain the parent array ` `static` `void` `dfs1(``int` `u, ``int` `count) ` `{ ` `    ``visited[u] = ``true``; ` `    ``int` `temp = ``0``; ` `    ``for` `(``int` `i = ``0``; i < adjacent.get(u).size(); i++) ` `    ``{ ` `        ``if` `(!visited[adjacent.get(u).get(i)])  ` `        ``{ ` `            ``temp++; ` `            ``parent[adjacent.get(u).get(i)] = u; ` `            ``dfs1(adjacent.get(u).get(i), count + ``1``); ` `        ``} ` `    ``} ` ` `  `    ``if` `(temp == ``0``) ` `    ``{ ` `        ``if` `(maxi < count)  ` `        ``{ ` `            ``maxi = count; ` `            ``endnode = u; ` `        ``} ` `    ``} ` `} ` ` `  `// DFS function to find the end node ` `// of the Longest Branch to Diameter ` `static` `void` `dfs2(``int` `u, ``int` `count) ` `{ ` `    ``visited[u] = ``true``; ` `    ``int` `temp = ``0``; ` `    ``for` `(``int` `i = ``0``; i < adjacent.get(u).size(); i++)  ` `    ``{ ` `        ``if` `(!visited[adjacent.get(u).get(i)] &&  ` `            ``!vis[adjacent.get(u).get(i)])  ` `        ``{ ` `            ``temp++; ` `            ``dfs2(adjacent.get(u).get(i), count + ``1``); ` `        ``} ` `    ``} ` `    ``if` `(temp == ``0``)  ` `    ``{ ` `        ``if` `(maxi < count)  ` `        ``{ ` `            ``maxi = count; ` `            ``thirdnode = u; ` `        ``} ` `    ``} ` `} ` ` `  `// Function to find the required nodes ` `static` `void` `findNodes() ` `{ ` `    ``// To find start node of diameter ` `    ``dfs(``1``, ``0``); ` ` `  `    ``for` `(``int` `i = ``0``; i <= N; i++) ` `        ``visited[i] = ``false``; ` ` `  `    ``maxi = -``1``; ` ` `  `    ``// To find end node of diameter ` `    ``dfs1(startnode, ``0``); ` ` `  `    ``for` `(``int` `i = ``0``; i <= N; i++) ` `        ``visited[i] = ``false``; ` ` `  `    ``// x is the end node of diameter ` `    ``int` `x = endnode; ` `    ``vis[startnode] = ``true``; ` ` `  `    ``// Mark all the nodes on diameter ` `    ``// using back tracking ` `    ``while` `(x != startnode)  ` `    ``{ ` `        ``vis[x] = ``true``; ` `        ``x = parent[x]; ` `    ``} ` ` `  `    ``maxi = -``1``; ` ` `  `    ``// Find the end node of longest ` `    ``// branch to diameter ` `    ``for` `(``int` `i = ``1``; i <= N; i++)  ` `    ``{ ` `        ``if` `(vis[i]) ` `            ``dfs2(i, ``0``); ` `    ``} ` `} ` ` `  `// Driver code ` `public` `static` `void` `main(String args[]) ` `{ ` `    ``for``(``int` `i = ``0``; i < MAX; i++) ` `    ``adjacent.add(``new` `Vector()); ` `     `  `    ``N = ``4``; ` `    ``adjacent.get(``1``).add(``2``); ` `    ``adjacent.get(``2``).add(``1``); ` `    ``adjacent.get(``1``).add(``3``); ` `    ``adjacent.get(``3``).add(``1``); ` `    ``adjacent.get(``1``).add(``4``); ` `    ``adjacent.get(``4``).add(``1``); ` ` `  `    ``findNodes(); ` ` `  `    ``System.out.print( ``"("` `+ startnode + ``", "` `+  ` `                            ``endnode + ``", "` `+  ` `                            ``thirdnode + ``")"``); ` `} ` `} ` ` `  `// This code is contributed by Arnab Kundu `

## Python3

 `# Python3 implementation of the approach ` ` `  `MAX` `=` `100005` `adjacent ``=` `[[] ``for` `i ``in` `range``(``MAX``)] ` `visited ``=` `[``False``] ``*` `MAX` `  `  `# To store the required nodes ` `startnode ``=` `endnode ``=` `thirdnode ``=` `None` `maxi, N ``=` `-``1``, ``None` `  `  `# Parent array to retrace the nodes ` `parent ``=` `[``None``] ``*` `MAX` `  `  `# Visited array to prevent DFS ` `# in direction on Diameter path ` `vis ``=` `[``False``] ``*` `MAX` `  `  `# DFS function to find the startnode ` `def` `dfs(u, count): ` ` `  `    ``visited[u] ``=` `True` `    ``temp ``=` `0` `    ``global` `startnode, maxi ` `     `  `    ``for` `i ``in` `range``(``0``, ``len``(adjacent[u])):  ` `        ``if` `not` `visited[adjacent[u][i]]:  ` `            ``temp ``+``=` `1` `            ``dfs(adjacent[u][i], count ``+` `1``) ` `  `  `    ``if` `temp ``=``=` `0``: ` `        ``if` `maxi < count: ` `            ``maxi ``=` `count ` `            ``startnode ``=` `u ` `  `  `# DFS function to find the endnode of ` `# diameter and maintain the parent array ` `def` `dfs1(u, count): ` ` `  `    ``visited[u] ``=` `True` `    ``temp ``=` `0` `    ``global` `endnode, maxi ` `     `  `    ``for` `i ``in` `range``(``0``, ``len``(adjacent[u])):  ` `        ``if` `not` `visited[adjacent[u][i]]: ` `            ``temp ``+``=` `1` `            ``parent[adjacent[u][i]] ``=` `u ` `            ``dfs1(adjacent[u][i], count ``+` `1``) ` `  `  `    ``if` `temp ``=``=` `0``: ` `        ``if` `maxi < count:  ` `            ``maxi ``=` `count ` `            ``endnode ``=` `u ` `         `  `# DFS function to find the end node ` `# of the Longest Branch to Diameter ` `def` `dfs2(u, count): ` ` `  `    ``visited[u] ``=` `True` `    ``temp ``=` `0` `    ``global` `thirdnode, maxi ` `     `  `    ``for` `i ``in` `range``(``0``, ``len``(adjacent[u])):  ` `        ``if` `(``not` `visited[adjacent[u][i]] ``and` `            ``not` `vis[adjacent[u][i]]): ` `            ``temp ``+``=` `1` `            ``dfs2(adjacent[u][i], count ``+` `1``) ` `         `  `    ``if` `temp ``=``=` `0``: ` `        ``if` `maxi < count:  ` `            ``maxi ``=` `count ` `            ``thirdnode ``=` `u ` `  `  `# Function to find the required nodes ` `def` `findNodes(): ` ` `  `    ``# To find start node of diameter ` `    ``dfs(``1``, ``0``) ` `    ``global` `maxi ` `  `  `    ``for` `i ``in` `range``(``0``, N``+``1``): ` `        ``visited[i] ``=` `False` `  `  `    ``maxi ``=` `-``1` `  `  `    ``# To find end node of diameter ` `    ``dfs1(startnode, ``0``) ` `  `  `    ``for` `i ``in` `range``(``0``, N``+``1``): ` `        ``visited[i] ``=` `False` `  `  `    ``# x is the end node of diameter ` `    ``x ``=` `endnode ` `    ``vis[startnode] ``=` `True` `  `  `    ``# Mark all the nodes on diameter ` `    ``# using back tracking ` `    ``while` `x !``=` `startnode:  ` `        ``vis[x] ``=` `True` `        ``x ``=` `parent[x] ` `     `  `    ``maxi ``=` `-``1` `  `  `    ``# Find the end node of longest ` `    ``# branch to diameter ` `    ``for` `i ``in` `range``(``1``, N``+``1``):  ` `        ``if` `vis[i]: ` `            ``dfs2(i, ``0``) ` `  `  `# Driver code ` `if` `__name__ ``=``=` `"__main__"``: ` ` `  `    ``N ``=` `4` `    ``adjacent[``1``].append(``2``) ` `    ``adjacent[``2``].append(``1``) ` `    ``adjacent[``1``].append(``3``) ` `    ``adjacent[``3``].append(``1``) ` `    ``adjacent[``1``].append(``4``) ` `    ``adjacent[``4``].append(``1``) ` `  `  `    ``findNodes() ` `  `  `    ``print``(``"({}, {}, {})"``.``format``(startnode, endnode, thirdnode)) ` `  `  `# This code is contributed by Rituraj Jain `

Output:

```(2, 3, 4)
```

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