# Maximize the sum of products of the degrees between any two vertices of the tree

• Last Updated : 25 May, 2021

Given an integer N, the task is to construct a tree such that the sum of for all ordered pairs (u, v) is the maximum where u != v. Print the maximum possible sum.

Examples:

```Input: N = 4
Output: 26
1
/
2
/
3
/
4
For node 1, 1*2 + 1*2 + 1*1 = 5
For node 2, 2*1 + 2*2 + 2*1 = 8
For node 3, 2*1 + 2*2 + 2*1 = 8
For node 4, 1*1 + 1*2 + 1*2 = 5
Total sum = 5 + 8 + 8 + 5 = 26

Input: N = 6
Output: 82 ```

Approach: We know that sum of the degree of all nodes in a tree is (2 * N) – 2 where N is the number of nodes in the tree. As we have to maximize the sum, we have to minimize the number of leaf nodes, as the leaf nodes have the minimum degree among all the nodes of the tree and the tree will be of the form:

```      1
/
2
/
...
/
N```

where only the root and the only leaf node will have a degree of 1 and all the other nodes will have degree 2.

Below is the implementation of the above approach:

## C++

 `// C++ implementation of above approach``#include ``using` `namespace` `std;``#define ll long long int` `// Function to return the maximum possible sum``ll maxSum(``int` `N)``{``    ``ll ans = 0;` `    ``for` `(``int` `u = 1; u <= N; u++) {``        ``for` `(``int` `v = 1; v <= N; v++) {``            ``if` `(u == v)``                ``continue``;` `            ``// Initialize degree for node u to 2``            ``int` `degreeU = 2;` `            ``// If u is the leaf node or the root node``            ``if` `(u == 1 || u == N)``                ``degreeU = 1;` `            ``// Initialize degree for node v to 2``            ``int` `degreeV = 2;` `            ``// If v is the leaf node or the root node``            ``if` `(v == 1 || v == N)``                ``degreeV = 1;` `            ``// Update the sum``            ``ans += (degreeU * degreeV);``        ``}``    ``}` `    ``return` `ans;``}` `// Driver code``int` `main()``{``    ``int` `N = 6;``    ``cout << maxSum(N);``}`

## Java

 `// Java implementation of above approach``class` `GFG``{``    ` `// Function to return the maximum possible sum``static` `int` `maxSum(``int` `N)``{``    ``int` `ans = ``0``;` `    ``for` `(``int` `u = ``1``; u <= N; u++)``    ``{``        ``for` `(``int` `v = ``1``; v <= N; v++)``        ``{``            ``if` `(u == v)``                ``continue``;` `            ``// Initialize degree for node u to 2``            ``int` `degreeU = ``2``;` `            ``// If u is the leaf node or the root node``            ``if` `(u == ``1` `|| u == N)``                ``degreeU = ``1``;` `            ``// Initialize degree for node v to 2``            ``int` `degreeV = ``2``;` `            ``// If v is the leaf node or the root node``            ``if` `(v == ``1` `|| v == N)``                ``degreeV = ``1``;` `            ``// Update the sum``            ``ans += (degreeU * degreeV);``        ``}``    ``}` `    ``return` `ans;``}` `// Driver code``public` `static` `void` `main(String[] args)``{``    ``int` `N = ``6``;``    ``System.out.println(maxSum(N));``}``}` `// This code is contributed by Code_Mech`

## Python3

 `# Python3 implementation of above approach` `# Function to return the maximum possible sum``def` `maxSum(N) :``    ``ans ``=` `0``;` `    ``for` `u ``in` `range``(``1``, N ``+` `1``) :``        ``for` `v ``in` `range``(``1``, N ``+` `1``) :``            ``if` `(u ``=``=` `v) :``                ``continue``;` `            ``# Initialize degree for node u to 2``            ``degreeU ``=` `2``;` `            ``# If u is the leaf node or the root node``            ``if` `(u ``=``=` `1` `or` `u ``=``=` `N) :``                ``degreeU ``=` `1``;` `            ``# Initialize degree for node v to 2``            ``degreeV ``=` `2``;` `            ``# If v is the leaf node or the root node``            ``if` `(v ``=``=` `1` `or` `v ``=``=` `N) :``                ``degreeV ``=` `1``;` `            ``# Update the sum``            ``ans ``+``=` `(degreeU ``*` `degreeV);``            ` `    ``return` `ans;` `# Driver code``if` `__name__ ``=``=` `"__main__"` `:``    ` `    ``N ``=` `6``;``    ``print``(maxSum(N));` `# This code is contributed by Ryuga`

## C#

 `// C# implementation of above approach``using` `System;``class` `GFG``{``    ` `// Function to return the maximum possible sum``static` `int` `maxSum(``int` `N)``{``    ``int` `ans = 0;` `    ``for` `(``int` `u = 1; u <= N; u++)``    ``{``        ``for` `(``int` `v = 1; v <= N; v++)``        ``{``            ``if` `(u == v)``                ``continue``;` `            ``// Initialize degree for node u to 2``            ``int` `degreeU = 2;` `            ``// If u is the leaf node or the root node``            ``if` `(u == 1 || u == N)``                ``degreeU = 1;` `            ``// Initialize degree for node v to 2``            ``int` `degreeV = 2;` `            ``// If v is the leaf node or the root node``            ``if` `(v == 1 || v == N)``                ``degreeV = 1;` `            ``// Update the sum``            ``ans += (degreeU * degreeV);``        ``}``    ``}` `    ``return` `ans;``}` `// Driver code``static` `void` `Main()``{``    ``int` `N = 6;``    ``Console.WriteLine(maxSum(N));``}``}` `// This code is contributed by Chandan_jnu`

## PHP

 ``

## Javascript

 ``
Output:
`82`

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