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

\(\mathrm{T}(\mathrm{n})=\sum_{\mathrm{k}=1}^{\mathrm{n}} T(K-1) T(x)\) where x is :

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  1. n − k + 1
  2. n − k
  3. n − k − 1
  4. n − k − 2 

Answer (Detailed Solution Below)

Option 1 : n − k + 1
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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-point-imageKey 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-imageAdditional 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.

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-> NIELIT Scientific Assistant city intimation slip 2025 has been released at the official website.

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