Deck 11: A: Trees

Question 1

fill in the blanks.
If each edge of has weight 1, then the cost of any spanning tree of minimum cost is .

Answer 1

15

Question 2

fill in the blanks.
If T is a full binary tree with 101 vertices, its maximum height is .

Answer 2

50

Question 3

fill in the blanks.
There are full binary trees with six vertices.

Answer 3

0

Question 4

fill in the blanks.
The bubble sort has complexity O( ).

Question 5

fill in the blanks.
If T is a full binary tree with 101 vertices, its minimum height is .

Question 6

fill in the blanks.
If T is a full binary tree with 50 leaves, its minimum height is .

Question 7

fill in the blanks.
If T is a full binary tree with 50 internal vertices, then T has vertices.

Question 8

fill in the blanks.
There are non-isomorphic rooted trees with four vertices.

Question 9

fill in the blanks.
If T is a binary tree with 100 vertices, its minimum height is .

Question 10

fill in the blanks.
If T is a tree with 999 vertices, then T has edges.

Question 11

fill in the blanks.
Every full binary tree with 61 vertices has leaves.

Question 12

fill in the blanks.
The best comparison-based sorting algorithms for a list of n items have complexity O( ).

Question 13

fill in the blanks.
The value of the arithmetic expression whose prefix representation is − 5 / · 6 2 − 5 3 is .

Question 14

fill in the blanks.
Write 3n − (k + 5) in prefix notation: .

Question 15

fill in the blanks.
C7 has spanning trees.

Question 16

fill in the blanks.
Every 3-ary tree with 13 vertices has leaves.

Question 17

fill in the blanks.
Every full binary tree with 50 leaves has vertices.

Question 18

fill in the blanks.
There are non-isomorphic trees with four vertices.

Question 19

fill in the blanks.
The minimum number of weighings with a pan balance scale needed to guarantee that you find the single counterfeit coin and determine whether it is heavier or lighter than the other coins in a group of five coins is .

Question 20

fill in the blanks.
If T is a full binary tree of height h, then the minimum number of leaves in T is and the maximum number of leaves in T is .

Question 21

Prove that if T is a full m-ary tree with l leaves, then T has (ml − 1)/(m − 1) vertices.

Question 22

Draw all nonisomorphic rooted trees with 4 vertices.

Question 23

fill in the blanks.
Every tree is bipartite.

Question 24

fill in the blanks.
There is a tree with degrees 3, 3, 2, 2, 1, 1, 1, 1.

Question 25

fill in the blanks.
There is a tree with degrees 3, 2, 2, 2, 1, 1, 1, 1, 1.

Question 26

fill in the blanks.
Every full 3-ary tree of height 2 has at least vertices and at most vertices.

Question 27

fill in the blanks.
If T is a tree with 17 vertices, then there is a simple path in T of length 17.

Question 28

fill in the blanks.
If T is a tree, then its vertex-chromatic number is and its region-chromatic number is .

Question 29

Suppose T is a full m-ary tree with l leaves. Prove that T has (l − 1)/(m − 1) internal vertices.

Question 30

fill in the blanks.
If two trees have the same number of vertices and the same degrees, then the two trees are isomorphic.

Question 31

fill in the blanks.
The largest number of leaves in a binary tree of height 5 is .

Question 32

fill in the blanks.
If T is a tree with 50 vertices, the largest degree that any vertex can have is 49.

Question 33

fill in the blanks.
There are full 3-ary trees with 6 vertices.

Question 34

fill in the blanks.
No tree has a Hamilton path.

Question 35

fill in the blanks.
Every full binary tree with 45 vertices has internal vertices.

Question 36

fill in the blanks.
In a binary tree with 16 vertices, there must be a path of length 4.

Question 37

fill in the blanks.
Every tree is planar.

Question 38

Suppose T is a full m-ary tree with i internal vertices. Prove that T has 1 + (m − 1)i leaves.

Question 39

Draw all nonisomorphic trees with 5 vertices.

Question 40

fill in the blanks.
A full 3-ary tree with 13 internal vertices has vertices.

Question 41

(a) Set up a binary tree for the following list, in the given order, using alphabetical ordering: SHE, SELLS, SEA, SHELLS, BY, THE, SEASHORE. (b) How many comparisons with words in the tree are needed to determine if the word SHARK is in the tree? (c) How many comparisons with words in the tree are needed to determine if the word SEAWEED is in the tree? (d) How many comparisons with words in the tree are needed to determine if the word SHELLS is in the tree?

Question 42

(a) Set up a binary tree for the following list, in the given order, using alphabetical ordering: STOP, LET, THERE, TAPE, NONE, YOU, ANT, NINE, OAT, NUT. (b) Explain step by step how you would search for the word TEST in your tree. (c) What is the height of the shortest binary search tree that can hold all 10 words? (d) Write the preorder traversal of the tree. (e) Write the postorder traversal of the tree. (f) Write the inorder traversal of the tree.

Question 43

Find the inorder traversal. Questions 53-55 refer to the tree at the right.

Question 44

Find the preorder traversal.

Question 45

Write the compound proposition in postfix notation.

Question 46

Prove that if T is a full m-ary tree with v vertices, then T has ((m − 1)v + 1)/m leaves.

Question 47

Find the inorder traversal of the parsing tree for

Question 48

Suppose that the universal address set address of a vertex v in an ordered rooted tree is 3.2.5.1.5. Find (a) the level of v. (b) the minimum number of siblings of v. (c) the address of the parent of v. (d) the minimum number of vertices in the tree.

Question 49

Draw a parsing tree for (a − (3 + 2b))/(c2 +
d).

Question 50

The string is postfix notation for an algebraic expression. Write the expression in prefix notation.

Question 51

The algebraic expression is written in prefix notation. Write the expression in postfix notation.

Question 52

Write the compound proposition in prefix notation.

Question 53

Suppose you have 5 coins, one of which is heavier than the other four. Draw the decision tree for using a pan balance scale to find the heavy coin.

Question 54

Write the compound proposition in infix notation.

Question 55

Suppose you have 50 coins, one of which is counterfeit (either heavier or lighter than the others). You use a pan balance scale to find the bad coin. Prove that 4 weighings are not enough to guarantee that you find the bad coin and determine whether it is heavier or lighter than the other coins.

Question 56

Find the postorder traversal of the parsing tree for

Question 57

Find the postorder traversal.

Question 58

Suppose you have 5 coins, one of which is counterfeit (either heavier or lighter than the other four). You use a pan balance scale to find the bad coin and determine whether it is heavier or lighter. (a) Prove that 2 weighings are not enough to guarantee that you find the bad coin and determine whether it is heavier or lighter. (b) Draw a decision tree for weighing the coins to determine the bad coin (and whether it is heavier or lighter) in the minimum number of weighings.

Question 59

Find the preorder traversal of the parsing tree for

Question 60

The string is postfix notation for an algebraic expression. Write the expression in infix notation.

Question 61

Using alphabetical ordering, find a spanning tree for this graph by using a breadth-first search.

Question 62

Find the value of (in prefix notation) if Questions 66-73 refer to this graph.

Question 63

Using alphabetical ordering, find a spanning tree for this graph by using a depth-first search.

Question 64

The string is prefix notation for an algebraic expression. Write the expression in postfix notation.

Question 65

The string is postfix notation for a logic expression; however, there is a misprint. The triangle should be one of these three: Determine which of these three it must be and explain your reasoning.

Question 66

Using reverse alphabetical ordering, find a spanning tree for the graph by using a depth-first search.

Question 67

The string is prefix notation for an algebraic expression. Write the expression in infix notation.

Question 68

Find a spanning tree for the graph K3,4 using a depth-first search. (Assume that the vertices are labeled u1, u2, u3 in one set and v1, v2, v3, v4 in the other set, and that alphabetical ordering is used in the search, with numerical ordering on the subscripts used to break ties.)

Question 69

Using the ordering B, G, J, A, C, I, F, H, D, E, find a spanning tree for this graph by using a depth-first search.

Question 70

Using the ordering C, D, E, F, G, H, I, J, A, B, C, find a spanning tree for this graph by using a breadth-first search.

Question 71

Using alphabetical ordering, find a spanning tree for this graph by using a depth-first search.

Question 72

Using the ordering B, G, J, A, C, I, F, H, D, E, find a spanning tree for this graph by using a depth-first search.

Question 73

Using the ordering C, D, E, F, G, H, I, J, A, B, C, find a spanning tree for this graph by using a depth-first search.

Question 74

Using the ordering C, D, E, F, G, H, I, J, A, B, C, find a spanning tree for this graph by using a depth-first search.

Question 75

Using reverse alphabetical ordering, find a spanning tree for the graph by using a depth-first search.

Question 76

Using the ordering B, G, J, A, C, I, F, H, D, E, find a spanning tree for this graph by using a breadth-first search. Questions 74-81 refer to this graph.

Question 77

Using reverse alphabetical ordering, find a spanning tree for the graph by using a breadth-first search.

Question 78

Using the ordering C, D, E, F, G, H, I, J, A, B, C, find a spanning tree for this graph by using a breadth-first search.

Question 79

Using reverse alphabetical ordering, find a spanning tree for the graph by using a breadth-first search.

Question 80

Using the ordering B, G, J, A, C, I, F, H, D, E, find a spanning tree for this graph by using a breadth-first search.