 11.R.1RQ: a) Define a tree.________________b) Define a forest.
 11.R.2RQ: Can there be two different simple paths between the vertices of a t...
 11.R.3RQ: Give at least three examples of how trees are used in modeling.
 11.R.4RQ: a) Define a rooted tree and the root of such a tree._______________...
 11.R.6RQ: a) Define a full mary tree.________________b) How many vertices do...
 11.R.5RQ: a) How many edges does a tree with n vertices have?________________...
 11.R.7RQ: a) What is the height of a rooted tree?________________b) What is a...
 11.R.8RQ: a) What is a binary search tree?________________b) Describe an algo...
 11.R.9RQ: a) What is a prefix code?________________b) How can a prefix code b...
 11.R.10RQ: a) Define preorder, inorder, and postorder tree traversal._________...
 11.R.11RQ: a) Explain how to use preorder, inorder, and postorder traversals t...
 11.R.12RQ: Show that the number of comparisons used by a sorting algorithm to ...
 11.R.13RQ: a) Describe the Huffman coding algorithm for constructing an optima...
 11.R.15RQ: a) What is a spanning tree of a simple graph?________________b) Whi...
 11.R.14RQ: Draw the game tree for nim if the starting position consists of two...
 11.R.16RQ: a) Describe two different algorithms for finding a spanning tree in...
 11.R.17RQ: a) Explain how backtracking can be used to determine whether a simp...
 11.R.18RQ: a) What is a minimum spanning tree of a connected weighted graph?__...
 11.R.19RQ: a) Describe Kruskal’s algorithm and Prim’s algorithm for finding mi...
Solutions for Chapter 11.R: Discrete Mathematics and Its Applications 7th Edition
Full solutions for Discrete Mathematics and Its Applications  7th Edition
ISBN: 9780073383095
Solutions for Chapter 11.R
Get Full SolutionsChapter 11.R includes 19 full stepbystep solutions. Discrete Mathematics and Its Applications was written by and is associated to the ISBN: 9780073383095. Since 19 problems in chapter 11.R have been answered, more than 199311 students have viewed full stepbystep solutions from this chapter. This expansive textbook survival guide covers the following chapters and their solutions. This textbook survival guide was created for the textbook: Discrete Mathematics and Its Applications, edition: 7.

Adjacency matrix of a graph.
Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected). Adjacency matrix of a graph. Square matrix with aij = 1 when there is an edge from node i to node j; otherwise aij = O. A = AT when edges go both ways (undirected).

Associative Law (AB)C = A(BC).
Parentheses can be removed to leave ABC.

Change of basis matrix M.
The old basis vectors v j are combinations L mij Wi of the new basis vectors. The coordinates of CI VI + ... + cnvn = dl wI + ... + dn Wn are related by d = M c. (For n = 2 set VI = mll WI +m21 W2, V2 = m12WI +m22w2.)

Exponential eAt = I + At + (At)2 12! + ...
has derivative AeAt; eAt u(O) solves u' = Au.

Fundamental Theorem.
The nullspace N (A) and row space C (AT) are orthogonal complements in Rn(perpendicular from Ax = 0 with dimensions rand n  r). Applied to AT, the column space C(A) is the orthogonal complement of N(AT) in Rm.

Hankel matrix H.
Constant along each antidiagonal; hij depends on i + j.

Hilbert matrix hilb(n).
Entries HU = 1/(i + j 1) = Jd X i 1 xj1dx. Positive definite but extremely small Amin and large condition number: H is illconditioned.

Jordan form 1 = M 1 AM.
If A has s independent eigenvectors, its "generalized" eigenvector matrix M gives 1 = diag(lt, ... , 1s). The block his Akh +Nk where Nk has 1 's on diagonall. Each block has one eigenvalue Ak and one eigenvector.

lAII = l/lAI and IATI = IAI.
The big formula for det(A) has a sum of n! terms, the cofactor formula uses determinants of size n  1, volume of box = I det( A) I.

Left inverse A+.
If A has full column rank n, then A+ = (AT A)I AT has A+ A = In.

Length II x II.
Square root of x T x (Pythagoras in n dimensions).

Lucas numbers
Ln = 2,J, 3, 4, ... satisfy Ln = L n l +Ln 2 = A1 +A~, with AI, A2 = (1 ± /5)/2 from the Fibonacci matrix U~]' Compare Lo = 2 with Fo = O.

Norm
IIA II. The ".e 2 norm" of A is the maximum ratio II Ax II/l1x II = O"max· Then II Ax II < IIAllllxll and IIABII < IIAIIIIBII and IIA + BII < IIAII + IIBII. Frobenius norm IIAII} = L La~. The.e 1 and.e oo norms are largest column and row sums of laij I.

Orthogonal subspaces.
Every v in V is orthogonal to every w in W.

Pivot.
The diagonal entry (first nonzero) at the time when a row is used in elimination.

Rank one matrix A = uvT f=. O.
Column and row spaces = lines cu and cv.

Row space C (AT) = all combinations of rows of A.
Column vectors by convention.

Singular Value Decomposition
(SVD) A = U:E VT = (orthogonal) ( diag)( orthogonal) First r columns of U and V are orthonormal bases of C (A) and C (AT), AVi = O'iUi with singular value O'i > O. Last columns are orthonormal bases of nullspaces.

Triangle inequality II u + v II < II u II + II v II.
For matrix norms II A + B II < II A II + II B II·

Vector v in Rn.
Sequence of n real numbers v = (VI, ... , Vn) = point in Rn.