It is graph theory question. Don't know how to solve it.

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. Fir instance, there are 2 binary trees in two vertices and 5 binary trees in three vertices, depicted in Figure 4. Let n E N be a natural number. . . . . 1 2 (a) Show that the number of binary trees 1n n vertices is CT, = 1:71 ( 7:\"). Hint: Notice that the numbers 0,, 2 g?) satisfy the recursion C'inFl _Z 0' 2071 i- i=0 (b) Establish a bijection between binary trees in n vertices and triangulations of regular convex (n + 2)-g0n. How many triangulations does an octagon have ? /\\,/\\/\\ FIGURE 4. The two rooted binary trees with two vertices (Left) )and the ve rooted binary trees with three vertices (Right)