23.1. Discrete Mathematics

Try these first

1. 1.

   How many full binary trees can be formed with 2 leaves?

   Hint

   Consider the definition of a full binary tree.

2. 2.

   What is a valid path from (2,2) to (0,0) in a grid?

   Hint

   Think about how many times you will move right and upward.

3. 3.

   How many leaves does a full binary tree with 3 internal nodes have?

   • 3
   • 4
   • 5

   Hint

   Think about how the structure of full binary trees works.

4. 4.

   True or False: All vertices of a polygon can form diagonals.

   • True
   • False

   Hint

   Consider the definition of what a diagonal is.

5. 5.

   Formulate and prove a relationship between the number of full binary trees and the Catalan sequence.

   Hint

   Consider how binary trees can represent multiplication.

6. 6.

   Provide a rigorous mathematical proof for the formula calculating diagonals in a polygon.

   Hint

   Break down sides and verify with smaller polygons.