23.2.1. Counting Valid Paths

Try these first

1. 1.

   What is a full binary tree?

   Hint

   Think about how many children each node should have.

2. 2.

   How do you calculate the number of diagonals in a triangle?

   Hint

   Consider what diagonal connects non-adjacent vertices.

3. 3.

   What characterizes a full binary tree?

   • Each node has either one or two children.
   • Each node has two or no children.
   • Nodes can have three children.

   Hint

   Consider what makes a tree 'full'.

4. 4.

   A valid path in a grid can involve which moves?

   • True
   • False

   Hint

   Think about the movement constraints.

5. 5.

   How many distinct full binary trees can you create with 5 leaves? Hint: Relate to Catalan numbers.

   Hint

   Remember to look up the Catalan number for n = 4.

6. 6.

   If there are 10 sides in a convex polygon, how many diagonals exist?

   Hint

   Calculate each step carefully and remember the formula!