Problems 12.4 page 306 in the textbook introduction to algorithms by cormen 3rd

Question

Problems 12.4 page 306 in the textbook introduction to algorithms by cormen 3rd edition
Quest)
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 = 1 and that, for n = 1,

bn=Sbk.bn-1-k for sigma running k=0 to n-1



b. Referring to Problem 4-4 for the definition of a generating function, let B(x) be the
generating function
Show that B(x) = xB(x)2 + 1, and hence one way to express B(x) in closed form is

B(X)=1/2x(1-v(1-4x))


The Taylor expansion of f(x) around the point x = a is given by

f(x)=(Sfk(a)/factorial(k)) * (x-a)k


where f(k)(x) is the kth derivative of f evaluated at x.
c. Show that

bn=4n/(vtn3/2) *(1+O(1/n))


(the nth Catalan number) by using the Taylor expansion of around x = 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 to be n(n - 1) (n - k + 1)/k! if k = 0, and 0 otherwise.)

Explanation / Answer

http://www.4shared.com/document/1ESbxegf/Introduction_to_Algorithms-Cor.htm is link for the solution manual for this book . so you can get that answer from that solution manual..

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