Let T(n) be the number of different binary search trees on n distinct elements-then

 where x is :

Let T(n) be the number of different binary search trees on n distinct elements. Then:

T(n) = ∑_k=1ⁿ T(k-1) T(x) where x is:

• 1) n - k + 1
• 2) n - k
• 3) n - k - 1
• 4) n - k - 2

The correct answer is option 1: n - k + 1

Key Points

• The formula for the number of different binary search trees (BSTs) on n distinct elements can be understood as follows:
  • For each element k (from 1 to n) chosen as the root, there are T(k-1) ways to arrange the elements to the left of k (i.e., the left subtree) and T(x) ways to arrange the elements to the right of k (i.e., the right subtree).
  • In this context, x represents the number of elements remaining after choosing k and the elements to its left, which is n - k + 1.
  • Thus, the formula is: T(n) = ∑_k=1ⁿ T(k-1) T(n - k + 1).

Additional Information

• The number of different BSTs on n distinct elements is also known as the nth Catalan number, which has a closed-form expression: C(n) = (1 / (n + 1)) (2n choose n).
• This counting problem is fundamental in combinatorial mathematics and has applications in various fields, including computer science and algorithm design.
• Understanding the properties and formulas of BSTs helps in optimizing search and sort operations in data structures.