Number of trees whose sum of degrees of all the vertices is L
Given an integer L which is the sum of degrees of all the vertices of some tree. The task is to find the count of all such distinct trees (labeled trees). Two trees are distinct if they have at least a single different edge.
Examples:
Input: L = 2
Output: 1
Input: L = 6
Output: 16
Simple Solution: A simple solution is to find the number of nodes of the tree which has sum of degrees of all vertices as L. Number of nodes in such a tree is n = (L / 2 + 1) as described in this article.
Now the solution is to form all the labeled trees which can be formed using n nodes. This approach is quite complex and for larger values of n it is not possible to find out the number of trees using this process.
Efficient Solution: An efficient solution is to find the number of nodes using Cayley’s formula which states that there are n(n – 2) trees with n labeled vertices. So the time complexity of the code now reduces to O(n) which can be further reduced to O(logn) using modular exponentiation.
Below is the implementation of the above approach:
C++
// C++ implementation of the approach #include <iostream> using namespace std; #define ll long long int // Iterative Function to calculate (x^y) in O(log y) ll power(int x, ll y) { // Initialize result ll res = 1; while (y > 0) { // If y is odd, multiply x with result if (y & 1) res = (res * x); // y must be even now // y = y / 2 y = y >> 1; x = (x * x); } return res; } // Function to return the count // of required trees ll solve(int L) { // number of nodes int n = L / 2 + 1; ll ans = power(n, n - 2); // Return the result return ans; } // Driver code int main() { int L = 6; cout << solve(L); return 0; } |
chevron_right
filter_none
Java
// Java implementation of the approach import java.io.*; class GFG { // Iterative Function to calculate (x^y) in O(log y) static long power(int x, long y) { // Initialize result long res = 1; while (y > 0) { // If y is odd, multiply x with result if (y==1) res = (res * x); // y must be even now // y = y / 2 y = y >> 1; x = (x * x); } return res; } // Function to return the count // of required trees static long solve(int L) { // number of nodes int n = L / 2 + 1; long ans = power(n, n - 2); // Return the result return ans; } // Driver code public static void main (String[] args) { int L = 6; System.out.println (solve(L)); } } // This code is contributed by ajit. |
chevron_right
filter_none
Python3
# Python implementation of the approach # Iterative Function to calculate (x^y) in O(log y) def power(x, y): # Initialize result res = 1; while (y > 0): # If y is odd, multiply x with result if (y %2== 1): res = (res * x); # y must be even now #y = y / 2 y = int(y) >> 1; x = (x * x); return res; # Function to return the count # of required trees def solve(L): # number of nodes n = L / 2 + 1; ans = power(n, n - 2); # Return the result return int(ans); L = 6; print(solve(L)); # This code has been contributed by 29AjayKumar |
chevron_right
filter_none
C#
// C# implementation of the approach using System; class GFG { // Iterative Function to calculate (x^y) in O(log y) static long power(int x, long y) { // Initialize result long res = 1; while (y > 0) { // If y is odd, multiply x with result if (y == 1) res = (res * x); // y must be even now // y = y / 2 y = y >> 1; x = (x * x); } return res; } // Function to return the count // of required trees static long solve(int L) { // number of nodes int n = L / 2 + 1; long ans = power(n, n - 2); // Return the result return ans; } // Driver code static public void Main () { int L = 6; Console.WriteLine(solve(L)); } } // This code is contributed by Tushil. |
chevron_right
filter_none
Output:
16
GeeksforGeeks has prepared a complete interview preparation course with premium videos, theory, practice problems, TA support and many more features. Please refer Placement 100 for details
Recommended Posts:
- Finding in and out degrees of all vertices in a graph
- Construct a graph from given degrees of all vertices
- Maximize the sum of products of the degrees between any two vertices of the tree
- Check whether given degrees of vertices represent a Graph or Tree
- Maximize the number of uncolored vertices appearing along the path from root vertex and the colored vertices
- Find K vertices in the graph which are connected to at least one of remaining vertices
- Minimum number of edges between two vertices of a graph using DFS
- Number of Simple Graph with N Vertices and M Edges
- Minimum number of edges between two vertices of a Graph
- Find maximum number of edge disjoint paths between two vertices
- Find the number of distinct pairs of vertices which have a distance of exactly k in a tree
- Number of occurrences of a given angle formed using 3 vertices of a n-sided regular polygon
- Calculate number of nodes between two vertices in an acyclic Graph by Disjoint Union method
- Number of triangles formed by joining vertices of n-sided polygon with one side common
- Number of ways a convex polygon of n+2 sides can split into triangles by connecting vertices
If you like GeeksforGeeks and would like to contribute, you can also write an article using contribute.geeksforgeeks.org or mail your article to contribute@geeksforgeeks.org. See your article appearing on the GeeksforGeeks main page and help other Geeks.
Please Improve this article if you find anything incorrect by clicking on the "Improve Article" button below.
- Find a Mother vertex in a Graph using Bit Masking
- Largest subarray sum of all connected components in undirected graph
- Convert undirected connected graph to strongly connected directed graph
- Minimum cost to traverse from one index to another in the String
- Minimum time required to fill the entire matrix with 1's