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Chapter 3: Problem 19

How many different binary search trees can be constructed using six distinct keys?

Short Answer

Expert verified
After making the calculations, the number of different binary search trees that can be constructed using six distinct keys is found to be 132.

Step by step solution

01

Understanding the basics of Binary Search Trees (BST)

A Binary Search Tree (BST) is a special form of binary tree where each node has a value, and all the nodes in the left subtree are less than the root, and all nodes in the right subtree are more than the root. When keys are distinct, the structure of the BST is influenced by the order in which the keys are inserted into the tree.
02

Understanding Catalan Numbers

The number of distinct Binary Search Trees can be calculated using the \(n^{th}\) Catalan number. The \(n^{th}\) Catalan number can be calculated by the formula: \[C_{n}=\frac{(2n)!}{(n+1)!n!}\]
03

Substitute value of n

Substitute \(n = 6\) into the Catalan number formula: \[C_{n}=\frac{(2*6)!}{(6+1)!6!}\]
04

Calculation

Calculate the numerator, denominator separately and then find the division.
05

Calculate Factorial using Stirling's approximation

Calculate Factorial using Stirling's approximation - \[n! \approx \sqrt{2\pi n} * \left(\frac{n}{e}\right)^n\]

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Most popular questions from this chapter

Implement Floyd's Algorithm for the Shortest Paths Problem 2 (Algorithm 3.4) on your system, and study its performance using different graphs.

Write a more detailed version of the dynamic programming algorithm for the Traveling Salesperson Problem (Algorithm 3.11 ).

Let us consider two sequences of characters \(S_{1}\) and \(S_{2}\), For example, we could have \(S_{1}=\) ASCMA \(^{*}\) MN and \(S_{2}=\) AXMC4ANR. Assuming that a subsequence of a sequence can be constructed by deleting any number of characters from any positions, use the dynamic programming approach to create an algorithm that finds the longest common subsequence of \(S_{1}\) and \(S_{2}\) This algorithm returns the maximum-length common subsequence of each sequence.

Use the dynamic programming approach to write an algorithm to find the maximum sum in any contiguous sublist of a given list of \(n\) real values. Analyze your algorithm, and show the results using order notation.

Generalize the Optimal Binary Search Tree algorithm (Algorithm 3.9 ) to the case where the search key may not be in the tree. That is, you should let \(q_{i}\) where \(i=0,1,2, \ldots, n,\) be the probability that a missing search key can be situated between \(K e y_{i}\) and \(K e y_{i+1}\). Analyze your generalized algorithm, and show the results using order notation.
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