Problem 6. (20 pts) A binary tree is a rooted tree in which each vertex has at most two children, which are ordered, and are referred to as the left child and the right child. For instance, there are 2 binary trees in two vertices and 5 binary trees in three vertices, depicted in Figure 4. Let n EN be a natural number. 1 (2n (a) Show that the number of binary trees in n vertices is Cn- Hint: Notice that the numbers Cn) satisfy the recursion (b) Establish a bijection between binary trees in n vertices and triangulations of regular convex (n + 2)-gon. How many triangulations does an octagon have? Figure 4. The two rooted binary trees with two vertices (Left) and the five rooted binary trees with three vertices (Right)