Deck 11: Trees

Question 1

There are full binary trees with six vertices.

Answer 1

0.

Question 2

If T is a full binary tree with 50 internal vertices, then T has vertices.

Answer 2

101.

Question 3

There are non-isomorphic rooted trees with four vertices.

Answer 3

4.

Question 4

If T is a binary tree with 100 vertices, its minimum height is .

Question 5

The bubble sort has complexity

Question 6

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 7

If T is a full binary tree with 101 vertices, its minimum height is .

Question 8

If T is a tree with 999 vertices, then T has edges.

Question 9

If each edge of

Question 10

The best comparison-based sorting algorithms for a list of n items have complexity

Question 11

There are non-isomorphic trees with four vertices.

Question 12

Write in prefix notation:

Question 13

Every 3-ary tree with 13 vertices has leaves.

Question 14

Every full binary tree with 50 leaves has vertices.

Question 15

If T is a full binary tree with 101 vertices, its maximum height is .

Question 16

If T is a full binary tree with 50 leaves, its minimum height is .

Question 17

The value of the arithmetic expression whose prefix representation is

Question 18

C₇ has, spanning trees.

Question 19

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 20

Every full binary tree with 61 vertices has leaves.

Question 21

mark the statement TRUE or FALSE.
Every tree is planar.

Question 22

There are full 3-ary trees with 6 vertices.

Question 23

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

Question 24

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

Question 25

mark the statement TRUE or FALSE.
If T is a tree with 50 vertices, the largest degree that any vertex can have is 49.

Question 26

A full 3-ary tree with 13 internal vertices has vertices.

Question 27

If T is a tree, then its vertex-chromatic number is and its region-chromatic number is .

Question 28

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

Question 29

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

Question 30

mark the statement TRUE or FALSE.
There is a tree with degrees 3, 3, 2, 2, 1, 1, 1, 1.

Question 31

Draw all nonisomorphic trees with 5 vertices.

Question 32

mark the statement TRUE or FALSE.
In a binary tree with 16 vertices, there must be a path of length 4.

Question 33

mark the statement TRUE or FALSE.
No tree has a Hamilton path.

Question 34

mark the statement TRUE or FALSE.
If T is a tree with 17 vertices, then there is a simple path in T of length 17.

Question 35

mark the statement TRUE or FALSE.
There is a tree with degrees 3, 2, 2, 2, 1, 1, 1, 1, 1.

Question 36

mark the statement TRUE or FALSE.
Every tree is bipartite.

Question 37

mark the statement TRUE or FALSE.
If T is a rooted binary tree of height 5, then T has at most 25 leaves.

Question 38

Draw all nonisomorphic rooted trees with 4 vertices.

Question 39

mark the statement TRUE or FALSE.
If two trees have the same number of vertices and the same degrees, then the two trees are isomorphic.

Question 40

Every full binary tree with 45 vertices has internal vertices.

Question 41

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 42

Find the preorder traversal of the parsing tree for

Question 43

Find the postorder traversal of the parsing tree for

Question 44

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 45

refer to the tree at the right.
Find the inorder traversal.

Question 46

Draw a parsing tree for

Question 47

(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 48

Write the compound proposition

Question 49

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 50

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

Question 51

Find the preorder traversal.

Question 52

refer to the tree at the right.
Find the postorder traversal.

Question 53

Find the inorder traversal of the parsing tree for

Question 54

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 55

Write the compound proposition

Question 56

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

Question 57

(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 58

Write the compound proposition

Question 59

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

Question 60

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

Question 61

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

Question 62

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

Question 63

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 64

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

Question 65

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

Question 66

refer to this graph.
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 67

refer to this graph.
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 68

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

Question 69

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

Question 70

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

Question 71

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

Question 72

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

Question 73

refer to this graph.
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

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

Question 75

refer to this graph.
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 76

refer to this graph.
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.

Question 77

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

Question 78

refer to this graph.
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 79

refer to this graph.
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 80

Find the value of (in prefix notation) if