Question

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

Short answer

Proof is based on the method of induction. It is shown that the formula holds for the base case \(n=0\), and assuming it's valid for \(n=k\), it's demonstrated to be true for \(n=k+1\) by applying the binomial coefficient properties and the induction hypothesis.

Step-by-step solution

Base Case

When \(n=0\), the formula becomes \(\frac{1}{1}\left(\begin{array}{l}0 \ 0\end{array}\right) = 1\), which is expected since there is only one way to form a binary search tree with no keys, that is, an empty tree.

Hypothesis

Assume the formula holds for \(n=k\). That means the number of binary search trees with \(k\) keys is given by the formula \(\frac{1}{(k+1)}\left(\begin{array}{l}2 k \ k\end{array}\right)\).

Inductive Step

Now, must prove it holds for \(n=k+1\). A binary search tree with \(k+1\) keys can be formed by choosing a root, forming a binary search tree with \(i\) keys on the left and a binary search tree with \(k-i\) keys on the right for a sum over all \(i\) from 0 to \(k\). Using the binomial coefficient properties and the induction hypothesis, derive that the formula holds for \(k+1\)-th case representing \(\frac{1}{(k+2)}\left(\begin{array}{l}2(k+1) \ (k+1)\end{array}\right)\) binary search trees.

Key concepts

Algorithm Analysis

Understanding the efficiency of an algorithm is critical in computer science. Algorithm analysis focuses on determining the amount of resources (such as time and space) required to execute a given algorithm. The goal is to classify algorithms according to their performance and resource consumption.

Binary search trees are a classic topic where algorithm analysis is applied. They store items in a sorted fashion, allowing for operations like search, insertion, and deletion to be conducted in logarithmic average time complexity, which is highly efficient. In the context of the given exercise, understanding algorithm analysis aids in appreciating why it is important to know the number of different binary search trees that can be formed with a given set of keys. The distinct structures of these trees can affect the performance of the operations on them.

When studying binary search trees, algorithm analysis helps us realize that not all trees are created equal. For example, a balanced tree can give us the best time complexity for operations (O(log n)), whereas a skewed tree (where all nodes are either to the left or right) would perform poorly (O(n)) as operations degrade to that of a linked list.

Combinatorics

Combinatorics, the field of mathematics concerned with counting, can take many forms such as combinations, permutations, and various arrangements. In the landscape of binary search trees (BSTs), combinatorics comes into play when we are interested in the number of different BSTs that can be constructed from a set of unique keys.

The problem from the textbook is a combinatorial problem because it involves counting the number of ways to arrange the keys into BSTs, where the arrangement has to satisfy the BST property (left subtree has smaller elements than the node, and the right subtree has larger elements). Combinatorics often involves establishing a systematic method for counting without having to list all possible scenarios, which is especially useful for larger values of n, where manually counting arrangements quickly becomes impractical.

Inductive Proofs

Inductive proofs are a powerful mathematical technique used to prove statements for all natural numbers. The process involves two critical steps: the base case and the inductive step.

For binary search trees, an inductive proof can demonstrate the formula for counting the number of trees. As shown in the provided solution, the base case is established when there are no keys, resulting in only one possible tree - an empty tree. Then, the inductive step assumes the formula is true for a tree with k keys and proves that if this is the case, it must also be true for a tree with k+1 keys. This proof style is not just about understanding a single case, but ensuring that the reasoning holds universally across all numbers of keys. It offers a concrete and rigorous method to confirm that the combinatorial formula for the number of possible binary search trees is indeed accurate.

Binomial Coefficients

Binomial coefficients, denoted as \( \binom{n}{k} \), represent the number of ways to choose k items from a set of n distinct items without considering the order of selection. They are a fundamental component of combinatorics and appear in the binomial theorem and Pascal's triangle.

In the context of our binary search tree problem, the binomial coefficient \( \binom{2n}{n} \) is used to count the number of ways to create BSTs with n keys. The additional \( \frac{1}{n+1} \) in the formula adjusts the count by considering the unique tree structures, since binomial coefficients alone would overcount the number of BSTs due to structural repetitions.

Understanding binomial coefficients allows us to manage and simplify complex combinatorial expressions, enabling us to derive and prove formulas such as the one used for the number of binary search trees, which would otherwise be daunting to handle.