Question

Show that the number of binary search trees with \(n\) keys is given by the formula \\[\frac{1}{(n+1)}\left(\begin{array}{c}2 n \\\n\end{array}\right)\\]

Short answer

The number of binary search trees with \(n\) keys follows the \(n^{th}\) Catalan Number, which is calculated by the formula \[\frac{1}{n+1}\binom{2n}{n} \]. This was proven by establishing a link between the recursive formula for the count of binary search trees and the recursive expression for Catalan Numbers.

Step-by-step solution

Understand the Problem

The problem is to prove the formula for the number of binary search trees possible with \(n\) keys. A binary search tree is a tree data structure in which each node has at most two children, referred to as the left child and the right child.

Find the Recursive Formula

For a BST, the number of total trees possible is the sum of the product of the number of trees possible in the left sub-tree and the right sub-tree. This can be represented with the recursive formula as: \[T(n) = \Sigma_{i=0}^{n-1} T(i) * T(n-i-1)\]

Link Recursive Form with Catalan Numbers

There is a pattern of numbers known as the Catalan Numbers. The \(n^{th}\) Catalan number is given by the formula: \[C(n) = \frac{1}{n+1} \binom{2n}{n}\] and the recursive formula for Catalan numbers is exactly the same as the formula derived in Step 2. This means that BST count is the same as Catalan numbers.

Conclusion

Then, from Step 2 and Step 3 it follows that the formula \[\frac{1}{(n+1)}\binom{2n}{n}\] is indeed the formula for calculating the number of distinct binary search trees with \(n\) keys.

Key concepts

Catalan Numbers

Catalan Numbers are a sequence of natural numbers that have deep mathematical significance. They frequently appear in various combinatorial mathematics problems. One fascinating property of Catalan numbers is their applicability in counting problems, such as the number of ways to correctly match parentheses, the number of rooted binary trees, and indeed, the number of binary search trees with a given number of nodes.

For any non-negative integer \(n\), the \(n^{th}\) Catalan number is computed using the formula:

• \[C(n) = \frac{1}{n+1} \binom{2n}{n}\]

This construction represents the number of ways to completely balance \(n\) pairs of parentheses. In the context of binary search trees, it describes how to arrange \(n\) keys while still preserving the BST properties.

• Allows for unique arrangements of elements
• Ensures correct structure and property retention in trees

Each Catalan number provides a count of how many BSTs can be established using a certain number of keys.

Recursive Formula

The recursive formula is a critical concept in computing, especially in algorithm design and analysis. It involves defining a sequence based on previous terms, which allows for complex problems to be broken down into simpler sub-problems. This methodology is key in calculating the number of possible binary search trees with \(n\) keys.

In the case of BSTs, the recursive formula is:

• \[T(n) = \sum_{i=0}^{n-1} T(i) * T(n-i-1)\]

This formula encapsulates the divide-and-conquer approach:

• Each tree has a root node, with potential sub-trees branching to the left and right.
• The number of BSTs can be determined by iterating over potential root nodes, creating sub-problems for left and right sub-trees.

By systematically evaluating smaller configurations, we recursively approach our solution, eventually mapping directly to Catalan numbers for optimized calculation.

Tree Data Structure

A tree data structure is a fundamental concept in computer science, used to represent hierarchical data. In a tree, each node can have zero or more child nodes, creating a wide array of possible structures. Binary trees are a special case of trees where each node has, at most, two children: left and right.

Binary Search Trees (BSTs), a subset of the binary tree, have a specific ordering property: for any given node, all values in its left subtree are less than the node's value, and all values in the right subtree are greater. This property makes BSTs valuable for fast data retrieval and efficient sorting.

• BSTs allow for easy data operations such as searching, insertion, and deletion.
• Trees are dynamic and can grow and shrink as needed, which makes them suitable for dynamic applications.

Understanding the structure of BSTs is crucial for explaining how varying baseline arrangements influence the total quantity of unique trees that can be configured with \(n\) elements.