Number of different binary trees Let bn denote the number
Chapter 12, Problem 12-4(choose chapter or problem)
Number of different binary trees Let bn denote the number of different binary trees with n nodes. In this problem, you will find a formula for bn, as well as an asymptotic estimate. a. Show that b0 D 1 and that, for n 1, bn D Xn1 kD0 bkbn1k : b. Referring to 4-4 for the definition of a generating function, let B.x/ be the generating function B.x/ D X1 nD0 bnxn : Show that B.x/ D xB.x/2 C 1, and hence one way to express B.x/ in closed form is B.x/ D 1 2x 1 p 1 4x : T. The Taylor expansion of f .x/ around the point x D a is given by f .x/ D X1 kD0 f .k/.a/ k .x a/k ; where f .k/.x/ is the kth derivative of f evaluated at x. c. Show that bn D 1 n C 1 2n n ! (the nth Catalan number) by using the Taylor expansion of p1 4x around x D 0. (If you wish, instead of using the Taylor expansion, you may use the generalization of the binomial expansion (C.4) to nonintegral exponents n, where for any real number n and for any integer k, we interpret n k to be n.n 1/.n k C 1/=k if k 0, and 0 otherwise.) d. Show that bn D 4n p n3=2 .1 C O.1=n// :
Unfortunately, we don't have that question answered yet. But you can get it answered in just 5 hours by Logging in or Becoming a subscriber.
Becoming a subscriber
Or look for another answer