2. Diagrams are constructed using the following procedure. First, a rectangle is drawn. Then exactly 17. points are marked along the top edge of the rectangle, and exactly 71 points are marked along the bottom edge of the rectangle. These points are then joined by curves such that 1. no curve can leave the rectangle; 2. no curve intersects any other curve; and 3. each of the 271. points is joined by a curve to exactly one other of the 2n points. A diagram following these rules is called a valid diagram with 2n marked points. Two valid diagrams are called equivalent if the 11. pairs of marked points connected by a curve are equal in both diagrams. Examples of valid diagrams for n = l, 2, 3 are shown below. C1 [1 [j Prove that the number of valid diagrams with 27?. marked points is equal to the nth Catalan number by (a) recovering the recurrence relation of the Catalan numbers from constructing valid diagrams with 27?. marked points from diagrams with fewer marked points. (b) constructing an explicit bijection between the set of valid diagrams with 2n marked points (up to equivalence) and another collection of objects you have studied in lectures that is known to be enumerated by the nth Catalan number. Show all your working