Prove the following using structural induction: For any rooted binary tree T the number of vertices |T| in T satisfies the inequality |T| ≤ (2^ (height(T)+1)) − 1.
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Prove the following using structural induction: For any rooted binary tree T the number of vertices |T| in T satisfies the inequality |T| ≤ (2^ (height(T)+1)) − 1.
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- 1a. Prove: every tree with n ≥ 2 vertices has at least 2 leaves. (3 pt) 1b. Let T be a tree. Prove: if all vertices have degree either 1 or at least 4, then T has at least 2(n + 1)/3 leaves. (4 pt)2 3) y=-(x– 4)³ + 2(x – 4)³ y Graph showing all algorithm and workLet > be a monomial order. If m_1 > m_2 > m_3 > ... is a decreasing sequence of monomials, prove that there exists a t such that m_t =m_{t+1} = ... (any decreasing sequence of monomials is eventually stationary)
- 5) Find The minimum value of The Funchun flad = 20²_n+²9 / prove that: Let G be agraph with n vertices, ny 2 و the h G has at Least two vertices which are not cut vertices.The graph of wheel denoted by W, is obtained when an additional vertex is added to cycle Cn, for n > 3, and connect this new vertex to each of n vertices by new edges. Match between each statement (a)- (d)) and a graph ((1)-(5)) such that the chosen graph satisfies the statement. (1) C, (2) C10 (3) W, (4) W, (5) W 10 A graph with the sum of degrees is 28. Choose... A simple and bipartite graph Choose... A simple graph with Hamiltonian circuit and vertex of degree 10. Choose... : A regular graph and not bipartite graph Choose...
- 4. The conventional algorithm for evaluating a polynomial anx" + an-1x¹ +... + a1x + ao at x = c can be expressed in pseudocode as Algorithm Polynomial (c, ao, a₁,...,an: real numbers) power = 1; y = ao; for i = 1 to n power power * c; y = y + ai * power return y; Notice that the final value of y is y = anc" + an-1 c¹ +...+ a₁c + ao, the value of the polynomial at x = c. a) Evaluate 3x² + x +1 at x = 2 by working through each step of the algorithm showing the values assigned at each assignment step. b) Exactly how many multiplications and additions are used to evaluate a polynomial of degree n at x = c? (Do not count additions used to increment the loop variable.)Suppose the following Branch-and-Bound tree is obtained for a pure integer maximization LP:Determine all pairs (x, y) of integers that satisfy x* + (x + 1)* = y² + (y + 1)²
- Let P₁ and P₂ be two paths of maximum length in a connected graph G. Prove that P₁ and P2 have a common vertex. Let G be a graph of order n and size strictly less than n - - 1. Prove that G is not connected.How many bijective functions have [10] as both the domain and codomain?ii) (Volterra set) Repeat the Cantor construction starting with the interval [0,1]. This time remove, in the k-iteration, an open interval of length 1/4k from the center of the remaining closed intervals to obtain SCV (4). ii-1) Draw the first 3 iterative steps of the construction of SCV (4) in detail. ii-2) Using the argument above or from Section 11.1 of the book (pages 330-332), compute the length of this Cantor-like set SCV (4), and prove that contains no intervals.
