PHP Program for nth Catalan Number
Last Updated :
09 Nov, 2023
Catalan numbers are defined as a mathematical sequence that consists of positive integers, which can be used to find the number of possibilities of various combinations.
Catalan numbers occur in many interesting counting problems like the following.
- Count the number of expressions containing n pairs of parentheses that are correctly matched. For n = 3, possible expressions are ((())), ()(()), ()()(), (())(), (()()).
- Count the number of possible Binary Search Trees with n keys (See this)
- Count the number of full binary trees (A rooted binary tree is full if every vertex has either two children or no children) with n+1 leaves.
- Given a number n, return the number of ways you can draw n chords in a circle with 2 x n points such that no 2 chords intersect.
Examples:
Input: n = 6
Output: 132
Input: n = 8
Output: 1430
PHP Program for nth Catalan Number using Recursive Solution:
Catalan numbers satisfy the following recursive formula: 
Step-by-step approach:
- Base condition for the recursive approach, when n <= 1, return 1
- Iterate from i = 0 to i < n
- Make a recursive call catalan(i) and catalan(n – i – 1) and keep adding the product of both into res.
- Return the res.
Below is the implementation of the above approach:
PHP
<?php
function catalan($n)
{
if ($n <= 1)
return 1;
$res = 0;
for($i = 0; $i < $n; $i++)
$res += catalan($i) *
catalan($n - $i - 1);
return $res;
}
for ($i = 0; $i < 10; $i++)
echo catalan($i), " ";
?>
|
Output1 1 2 5 14 42 132 429 1430 4862
Time Complexity:
PHP Program for nth Catalan Number using Dynamic Programming:
Step-by-step approach:
- Create an array catalan[] for storing ith Catalan number.
- Initialize, catalan[0] and catalan[1] = 1
- Loop through i = 2 to the given Catalan number n.
- Loop through j = 0 to j < i and Keep adding value of catalan[j] * catalan[i – j – 1] into catalan[i].
- Finally, return catalan[n]
Below is the implementation of the above approach:
PHP
<?php
function catalanDP( $n)
{
$catalan= array();
$catalan[0] = $catalan[1] = 1;
for ($i = 2; $i <= $n; $i++)
{
$catalan[$i] = 0;
for ( $j = 0; $j < $i; $j++)
$catalan[$i] += $catalan[$j] *
$catalan[$i - $j - 1];
}
return $catalan[$n];
}
for ($i = 0; $i < 10; $i++)
echo catalanDP($i) , " ";
?>
|
Output1 1 2 5 14 42 132 429 1430 4862
Time Complexity: O(n2)
Auxiliary Space: O(n)
Please refer to the complete article on Program for nth Catalan Number for more details!
PHP Program for nth Catalan Number using Binomial Coefficient:
We can also use the below formula to find nth Catalan number in O(n) time.
[Tex]C_n=\frac{1}{n+1}\binom{2n}{n}
[/Tex]
Below are the steps for calculating nCr.
- Create a variable to store the answer and change r to n – r if r is greater than n – r because we know that C(n, r) = C(n, n-r) if r > n – r
- Run a loop from 0 to r-1
- In every iteration update ans as (ans*(n-i))/(i+1), where i is the loop counter.
- So the answer will be equal to ((n/1)*((n-1)/2)*…*((n-r+1)/r), which is equal to nCr.
Below are steps to calculate Catalan numbers using the formula: 2nCn/(n+1)
- Calculate 2nCn using the similar steps that we use to calculate nCr
- Return the value 2nCn/ (n + 1)
Below is the implementation of the above approach:
PHP
<?php
function binomialCoeff($n, $k)
{
$res = 1;
if ($k > $n - $k)
$k = $n - $k;
for ($i = 0; $i < $k; ++$i)
{
$res *= ($n - $i);
$res = floor($res / ($i + 1));
}
return $res;
}
function catalan($n)
{
$c = binomialCoeff(2 * ($n), $n);
return floor($c / ($n + 1));
}
for ($i = 0; $i < 10; $i++)
echo catalan($i), " " ;
?>
|
Output1 1 2 5 14 42 132 429 1430 4862
Time Complexity: O(n).
Auxiliary Space: O(1)
We can also use the below formulas to find nth Catalan number in O(n) time.
[Tex]C_n=\frac{(2n)!}{(n+1)!n!}=\prod_{k=2}^{n}\frac{n+k}{k} \ for \ n\geq 0
[/Tex]
[Tex]C_n = \frac{2(2n-1)}{n+1}*C_{n-1} \ \ \ {|} \ \ {n>0}
[/Tex]
PHP Program for nth Catalan Number using the previous Catalan Number:
Below are steps to calculate Catalan numbers using the above formula:
- Initialize a variable res = 1
- Print 1 as the first Catalan Number
- Iterate from i = 1 to i < n
- Update res with res = (res * (4 * i – 2)) / (i + 1)
- print res
Below is the implementation for the above approach:
PHP
<?php
function catalan($n) {
$res = 1;
echo $res . " ";
for ($i = 1; $i < $n; $i++) {
$res = intval($res * (4 * $i - 2) / ($i + 1));
echo $res . " ";
}
}
$n = 10;
catalan($n);
?>
|
Output1 1 2 5 14 42 132 429 1430 4862
Time Complexity: O(n)
Auxiliary Space: O(1), since no extra space has been taken.
Please refer complete article on Program for nth Catalan Number for more details!
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