python The number of possible binary tree topologies (given all possible heights) with n nodes is the Catalan Number (in closed form) Cn = 1/(n+1)*(2n choose n). Recursively, this can also be written as: question ) Write a function cn_fast(n) that computes the n-th catalan number using the closed form formulation, and Conduct doubling experiments (just n = 1, 2, 4, and 8) to compute runtimes. Plot the result on a log-log plot.

Python Using recursion only No loops If the list is empty or does not contain any sets, the function should return 0. Using type() function and must be return not print def countSets(lst):

Write a recursive implementation of Euclid's Algorith for finding the greatest common divisor(GCD) of two intergers. Descriptions of this algorithm are available in algebra books and on theweb. (Note: A nonrecursive version of the GCD problem was given in the programming exercisesfor Chapter 7.) Write a test program that calls your GCD procedure five times, using thefollowing pairs of integers: (5,20),(24,18),(11,7),(432,226),(26,13). After each procedure call,display the GCD.

Consider a recursive function, called f, that computes powers of 3 using only the + operator. Assume n > = 0. int f(int n) { if (n == 0) return 1; return f(n-1) + f(n-1) + f(n-1); } Give an optimized version of f, called g, where we save the result of the recursive call to a temporary variable t, then return t+t+t. i got int g(int n) { if (n == 0) return 1; int t = g(n - 1); return t+t+t; } so now Write a recurrence relation for T(n), the number addition operations performed by g(n) in terms of n.

Solution for: Python

Given a list of integers, we want to know whether it is possible to choose a subset of some of the integers, such that the integers in the subset adds up to the given sum recursively. We also want that if an integer is chosen to be in the sum, the integer next to it in the list must be skipped and not chosen to be in the sum. Do not use any loops or regular expressions. Test cases: skipSum([2, 5, 10, 6], 12) true skipSum([2, 5, 10, 6], 7) false skipSum([2, 5, 10, 6], 16) false Given code: public static boolean skipSum (List list, int sum) { // call your recursive helper method return skipSumHelper (list, e, sum); 1. 2. 3. 4.

Write a recursive Python function named pgcd, to find and return the Greatest Common Divisor (GCD) of two numbers x and y passed in parameters. • Test it gcd (1234,4321) gcd (8192,192)