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Create number_of_possible_binary_trees.py
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algorithm/data_structures/binary_tree/number_of_possible_binary_trees.py
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""" | ||
Hey, we are going to find an exciting number called Catalan number which is use to find | ||
the number of possible binary search trees from tree of a given number of nodes. | ||
We will use the formula: t(n) = SUMMATION(i = 1 to n)t(i-1)t(n-i) | ||
Further details at Wikipedia: https://en.wikipedia.org/wiki/Catalan_number | ||
""" | ||
""" | ||
Our Contribution: | ||
Basically we Create the 2 function: | ||
1. catalan_number(node_count: int) -> int | ||
Returns the number of possible binary search trees for n nodes. | ||
2. binary_tree_count(node_count: int) -> int | ||
Returns the number of possible binary trees for n nodes. | ||
""" | ||
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def binomial_coefficient(n: int, k: int) -> int: | ||
""" | ||
Since Here we Find the Binomial Coefficient: | ||
https://en.wikipedia.org/wiki/Binomial_coefficient | ||
C(n,k) = n! / k!(n-k)! | ||
:param n: 2 times of Number of nodes | ||
:param k: Number of nodes | ||
:return: Integer Value | ||
>>> binomial_coefficient(4, 2) | ||
6 | ||
""" | ||
result = 1 # To kept the Calculated Value | ||
# Since C(n, k) = C(n, n-k) | ||
if k > (n - k): | ||
k = n - k | ||
# Calculate C(n,k) | ||
for i in range(k): | ||
result *= n - i | ||
result //= i + 1 | ||
return result | ||
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def catalan_number(node_count: int) -> int: | ||
""" | ||
We can find Catalan number many ways but here we use Binomial Coefficient because it | ||
does the job in O(n) | ||
return the Catalan number of n using 2nCn/(n+1). | ||
:param n: number of nodes | ||
:return: Catalan number of n nodes | ||
>>> catalan_number(5) | ||
42 | ||
>>> catalan_number(6) | ||
132 | ||
""" | ||
return binomial_coefficient(2 * node_count, node_count) // (node_count + 1) | ||
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def factorial(n: int) -> int: | ||
""" | ||
Return the factorial of a number. | ||
:param n: Number to find the Factorial of. | ||
:return: Factorial of n. | ||
>>> import math | ||
>>> all(factorial(i) == math.factorial(i) for i in range(10)) | ||
True | ||
>>> factorial(-5) # doctest: +ELLIPSIS | ||
Traceback (most recent call last): | ||
... | ||
ValueError: factorial() not defined for negative values | ||
""" | ||
if n < 0: | ||
raise ValueError("factorial() not defined for negative values") | ||
result = 1 | ||
for i in range(1, n + 1): | ||
result *= i | ||
return result | ||
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def binary_tree_count(node_count: int) -> int: | ||
""" | ||
Return the number of possible of binary trees. | ||
:param n: number of nodes | ||
:return: Number of possible binary trees | ||
>>> binary_tree_count(5) | ||
5040 | ||
>>> binary_tree_count(6) | ||
95040 | ||
""" | ||
return catalan_number(node_count) * factorial(node_count) | ||
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if __name__ == "__main__": | ||
node_count = int(input("Enter the number of nodes: ").strip() or 0) | ||
if node_count <= 0: | ||
raise ValueError("We need some nodes to work with.") | ||
print( | ||
f"Given {node_count} nodes, there are {binary_tree_count(node_count)} " | ||
f"binary trees and {catalan_number(node_count)} binary search trees." | ||
) |