Count Balanced Binary Trees of Height h

Given a height h, count and return the maximum number of balanced binary trees possible with height h. A balanced binary tree is one in which for every node, the difference between heights of left and right subtree is not more than 1.

Examples :

Input : h = 3
Output : 15

Input : h = 4
Output : 315

Recommended: Please solve it on “PRACTICE” first, before moving on to the solution.

Following are the balanced binary trees of height 3. Height of tree, h = 1 + max(left height, right height)
Since the difference between the heights of left and right subtree is not more than one, possible heights of left and right part can be one of the following:

1. (h-1), (h-2)
2. (h-2), (h-1)
3. (h-1), (h-1)
count(h) = count(h-1) * count(h-2) +
count(h-2) * count(h-1) +
count(h-1) * count(h-1)
= 2 * count(h-1) * count(h-2) +
count(h-1) * count(h-1)
= count(h-1) * (2*count(h - 2) +
count(h - 1))

Hence we can see that the problem has optimal substructure property.

A recursive function to count no of balanced binary trees of height h is:

int countBT(int h)
{
// One tree is possible with height 0 or 1
if (h == 0 || h == 1)
return 1;
return countBT(h-1) * (2 *countBT(h-2) +
countBT(h-1));
}

The time complexity of this recursive approach will be exponential. The recursion tree for the problem with h = 3 looks like : As we can see, sub-problems are solved repeatedly. Therefore we store the results as we compute them.
An efficient dynamic programming approach will be as follows :
Below is the implementation of above approach:

C++

 // C++ program to count number of balanced // binary trees of height h. #include #define mod 1000000007 using namespace std;     long long int countBT(int h) {             long long int dp[h + 1];     //base cases     dp = dp = 1;     for(int i = 2; i <= h; i++) {         dp[i] = (dp[i - 1] * ((2 * dp [i - 2])%mod + dp[i - 1])%mod) % mod;     }     return dp[h]; }       // Driver program int main() {     int h = 3;     cout << "No. of balanced binary trees"             " of height h is: "          << countBT(h) << endl; }

Java

 // Java program to count number of balanced  // binary trees of height h.  class GFG {            static final int MOD = 1000000007;            public static long countBT(int h) {         long[] dp = new long[h + 1];                    // base cases         dp = 1;         dp = 1;                    for(int i = 2; i <= h; ++i)              dp[i] = (dp[i - 1] * ((2 * dp [i - 2])% MOD + dp[i - 1]) % MOD) % MOD;                            return dp[h];     }            // Driver program     public static void main (String[] args) {         int h = 3;          System.out.println("No. of balanced binary trees of height "+h+" is: "+countBT(h));      } } /* This code is contributed by Brij Raj Kishore */

Python3

 # Python3 program to count number of balanced  # binary trees of height h.     def countBT(h) :     MOD = 1000000007     #initialize list     dp = [0 for i in range(h + 1)]            #base cases     dp = 1     dp = 1            for i in range(2, h + 1) :         dp[i] = (dp[i - 1] * ((2 * dp [i - 2])%MOD + dp[i - 1])%MOD) % MOD            return dp[h]    #Driver program h = 3 print("No. of balanced binary trees of height "+str(h)+" is: "+str(countBT(h)))    # This code is contributed by # Brij Raj Kishore

C#

 // C# program to count number of balanced  // binary trees of height h.     using System; class GFG {             static int MOD = 1000000007;             public static long countBT(int h) {          long[] dp = new long[h + 1];                     // base cases          dp = 1;          dp = 1;                     for(int i = 2; i <= h; ++i)              dp[i] = (dp[i - 1] * ((2 * dp [i - 2])% MOD + dp[i - 1]) % MOD) % MOD;                             return dp[h];      }             // Driver program      static void Main () {          int h = 3;          Console.WriteLine("No. of balanced binary trees of height "+h+" is: "+countBT(h));      }      // This code is contributed by Ryuga }

PHP



Output :

No of balanced binary trees of height h is: 15

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