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Enumeration of Binary Trees in C++
Enumeration of Binary Tree is counting the total number of distinct unlabeled binary trees of a given size (specific number of nodes). In this article, we will create a program to count the number of Binary Trees of n nodes.
Based on labeling of nodes of binary tree, it is of two types:
- Labeled Binary Tree
- Unlabeled Binary Tree
Labeled Binary Tree: It is a binary Tree in which the nodes of a tree are labeled with values.
Different Type of Labeled Binary Tree for a given number of nodes :
Number of nodes N = 2
Similarly, We can find the number of distinct labeled binary Tree for N number of nodes,
N = 1, count = 1
N = 2, count = 4
N = 3, count = 30
N = 4, count = 336
Here, we can see for each of the labeled nodes all the type of arrangements made for unlabeled nodes are made. So, the count will be n! * count of unlabeled binary trees.
C(N) = n! * ( (2n!) / (n+1)! * n! ) )
Program to illustrate the number of distinct unlabeled binary Tree for a given number of nobes N,
Example
#include <iostream>
using namespace std;
int fact(int n){
if(n == 1)
return 1;
return n * fact(n - 1);
}
int distinctCountLabeledTree(int N){
return ( (fact(N))*( fact(2*N) / ( fact(N+1)*fact(N)) ) ) ;
}
int main(){
int N = 6;
cout<<"The number of Distinct labeled Binary Tree is "<<distinctCountLabeledTree(N);
return 0;
}Output −
The number of Distinct labeled Binary Tree is 95040
Unlabeled Binary Tree: It is a binary Tree in which the nodes of a tree are not labeled with values .
Different type of Unlabeled binary Tree for a given number of nodes:
Number of Nodes N = 2
Number of distinct unlabeled binary Tree = 2
Similarly, We can find the number of distinct unlabeled binary trees for N.
N = 1, count = 1
N = 2, count = 2
N = 3, count = 5
N = 4, count = 14
Using this we can formulate the number of distinct unlabeled binary tree for N nodes,
It is given by Catalan number,
Another formula can be,
C(N) = (2n!) / (n+1)! * n!
Program to find the number of distinct unlabeled binary Tree for a given number of nodes N,
Example
#include <iostream>
using namespace std;
int fact(int n){
if(n == 1)
return 1;
return n * fact(n - 1);
}
int distinctCount(int N){
return ( fact(2*N) / ( fact(N+1)*fact(N) ) );
}
int main(){
int N = 7;
cout<<"The number of Distinct unlabeled Binary Tree is "<<distinctCount(N);
return 0;
}Output −
The number of Distinct unlabeled Binary Tree is 6
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