mirror of
https://github.com/TheAlgorithms/Python.git
synced 2024-11-24 13:31:07 +00:00
bc8df6de31
* [pre-commit.ci] pre-commit autoupdate updates: - [github.com/astral-sh/ruff-pre-commit: v0.2.2 → v0.3.2](https://github.com/astral-sh/ruff-pre-commit/compare/v0.2.2...v0.3.2) - [github.com/pre-commit/mirrors-mypy: v1.8.0 → v1.9.0](https://github.com/pre-commit/mirrors-mypy/compare/v1.8.0...v1.9.0) * [pre-commit.ci] auto fixes from pre-commit.com hooks for more information, see https://pre-commit.ci --------- Co-authored-by: pre-commit-ci[bot] <66853113+pre-commit-ci[bot]@users.noreply.github.com>
104 lines
2.8 KiB
Python
104 lines
2.8 KiB
Python
"""
|
|
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.
|
|
"""
|
|
|
|
|
|
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
|
|
|
|
|
|
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)
|
|
|
|
|
|
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
|
|
|
|
|
|
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)
|
|
|
|
|
|
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."
|
|
)
|