Discrete Mathematics

This section introduces key concepts in discrete mathematics, focusing on full binary trees, catalan numbers, valid paths in grids, triangulations of polygons, and derangements.

Full Binary Tree Definition

This section defines a full binary tree and explores its properties, particularly focusing on the structural characteristics and how they relate to the Catalan numbers.

Recurrence Relation For Full Binary Trees

This section discusses full binary trees and establishes a recurrence relation for counting structurally different full binary trees based on the number of leaves.

Bijection With Parenthesizing

This section introduces the concept of establishing a bijection between full binary trees with a certain number of leaves and ways of parenthesizing those leaves, explaining how this connects to the Catalan numbers.

Validity Of Paths In A Square Grid

This section discusses the number of valid paths in a square grid from the origin (0,0) to (n,n) using only upward and rightward movements.

Counting Valid Paths

This section explores how valid paths can be counted in contexts such as binary trees and grid movements, using combinatorial methods.

Counting Diagonals In Convex Polygons

This section explores the method to calculate the number of diagonals in a convex polygon based on its number of sides.

Finding Number Of Diagonals

This section discusses how to determine the number of diagonals in a convex polygon based on its number of sides using combinatorial methods.

Triangulations Of Convex Polygons

This section covers the concept of triangulations of convex polygons, exploring the relationship between the number of triangulations and the nth Catalan number.

Defining Triangulations

This section explores the concept of triangulations in polygons and their mathematical foundations.

Recurrence Relation For Triangulations

This section presents the recurrence relation for the number of triangulations of a convex polygon and establishes a connection to Catalan numbers.

Derangements Of N Objects

The section discusses the concept of derangements, exploring the categorization and recursive relationships involved in determining the number of derangements for a given set of n objects.

Categories Of Derangements

This section discusses the concept of derangements, defining them as permutations where none of the objects are in their original positions.

Overall Formula For Derangements

This section discusses the concept of derangements—permutations of n objects where no object appears in its original position—and derives the recurrence relation for calculating derangements.