Question 1

Find a minimal spanning tree for this weighted graph using Prim's algorithm.

Accepted Answer

In order, the following edges are added: $\{ d , e \} , \{ e , h \} , \{ e , f \} , \{ d , g \} , \{ g , a \} , \{ a , b \} , \{ b , c \} , \{ c , i \}$. The weight of the minimal spanning tree is 17 .

Question 2

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.

Accepted Answer

The answer of refer to this graph.  
-Using the...

Question 3

Is the following code a preﬁx code: A: 11, B: 10, C: 0?

Accepted Answer

Yes

Question 4

C7  has,   spanning trees.

Accepted Answer

Question 5

Draw all nonisomorphic rooted trees with 4 vertices.

Accepted Answer

The answer of Draw all nonisomorphic rooted trees with 4...

Question 6

Use the bubble sort to sort the list  5,4,3,2,1  in increasing order.

Accepted Answer

The answer of Use the bubble sort to sort the...

Question 7

Every full binary tree with 61 vertices has leaves.

Accepted Answer

The answer of Every full binary tree with 61 vertices...

Question 8

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

Accepted Answer

The answer of refer to this graph.  
-Using alphabetical...

Question 9

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

Accepted Answer

A) True 
 B)False

Question 10

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

Accepted Answer

The answer of Suppose T is a full m-ary tree...

Question 11

The string $$- \cdot 2 - x a + 4 y$$ is preﬁx notation for an algebraic expression. Write the expression in postﬁx
notation.

Accepted Answer

The answer of The string \[- \cdot 2 - x...

Question 12

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
.

Accepted Answer

The answer of The minimum number of weighings with a...

Question 13

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

Accepted Answer

The answer of refer to the tree at the right.
-Find...

Question 14

Write the compound proposition $$( \neg p ) \rightarrow ( q \vee ( r \wedge \neg s ) ) \text { in infix notation. }$$

Accepted Answer

The answer of Write the compound proposition \[( \neg p...

Question 15

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.

Accepted Answer

The answer of Suppose you have 5 coins, one of...

Question 16

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

Accepted Answer

The answer of Suppose T is a full m-ary tree...

Question 17

The string $$23 a \cdot x + 4 \uparrow + 7 \uparrow$$ is postﬁx notation for an algebraic expression. Write the expression in preﬁx
notation.

Accepted Answer

The answer of The string \[23 a \cdot x +...

Question 18

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

Accepted Answer

The answer of If T is a full binary tree...

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 .

Accepted Answer

The answer of If T is a full binary tree...

Question 20

Use backtracking to ﬁnd a sum of integers in the set $$\{ 18,19,23,25,31 \} \text { that equals } 44 \text {. }$$

Accepted Answer

The answer of Use backtracking to ﬁnd a sum of...

Find a minimal spanning tree for this weighted graph using Prim's algorithm.

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.

Is the following code a preﬁx code: A: 11, B: 10, C: 0?

C₇ has, spanning trees.

Draw all nonisomorphic rooted trees with 4 vertices.

Use the bubble sort to sort the list 5,4,3,2,1 in increasing order.

Every full binary tree with 61 vertices has leaves.

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

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

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

The string $- \cdot 2 - x a + 4 y$ is preﬁx notation for an algebraic expression. Write the expression in postﬁx notation.

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 .

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

Write the compound proposition $( \neg p ) \rightarrow ( q \vee ( r \wedge \neg s ) ) \text { in infix notation. }$

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.

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

The string $23 a \cdot x + 4 \uparrow + 7 \uparrow$ is postﬁx notation for an algebraic expression. Write the expression in preﬁx notation.

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

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 .

Use backtracking to ﬁnd a sum of integers in the set $\{ 18,19,23,25,31 \} \text { that equals } 44 \text {. }$

The Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Boolean Algebra

Modeling Computation

Filters

Exam 11: Trees

Find a minimal spanning tree for this weighted graph using Prim's algorithm.

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.

Is the following code a preﬁx code: A: 11, B: 10, C: 0?

C7 has, spanning trees.

Draw all nonisomorphic rooted trees with 4 vertices.

Use the bubble sort to sort the list 5,4,3,2,1 in increasing order.

Every full binary tree with 61 vertices has leaves.

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

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

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

The string −⋅2−xa+4y- \cdot 2 - x a + 4 y−⋅2−xa+4y is preﬁx notation for an algebraic expression. Write the expression in postﬁx notation.

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 .

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

Write the compound proposition (¬p)→(q∨(r∧¬s)) in infix notation. ( \neg p ) \rightarrow ( q \vee ( r \wedge \neg s ) ) \text { in infix notation. }(¬p)→(q∨(r∧¬s)) in infix notation.

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.

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

The string 23a⋅x+4↑+7↑23 a \cdot x + 4 \uparrow + 7 \uparrow23a⋅x+4↑+7↑ is postﬁx notation for an algebraic expression. Write the expression in preﬁx notation.

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

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 .

Use backtracking to ﬁnd a sum of integers in the set {18,19,23,25,31} that equals 44. \{ 18,19,23,25,31 \} \text { that equals } 44 \text {. }{18,19,23,25,31} that equals 44.

The Foundations: Logic and Proofs

Basic Structures: Sets, Functions, Sequences, Sums, Matrices

Algorithms

Number Theory and Cryptography

Induction and Recursion

Counting

Discrete Probability

Advanced Counting Techniques

Relations

Graphs

Boolean Algebra

Modeling Computation

Filters

C₇ has, spanning trees.

The string $- \cdot 2 - x a + 4 y$ is preﬁx notation for an algebraic expression. Write the expression in postﬁx notation.

Write the compound proposition $( \neg p ) \rightarrow ( q \vee ( r \wedge \neg s ) ) \text { in infix notation. }$

The string $23 a \cdot x + 4 \uparrow + 7 \uparrow$ is postﬁx notation for an algebraic expression. Write the expression in preﬁx notation.

Use backtracking to ﬁnd a sum of integers in the set $\{ 18,19,23,25,31 \} \text { that equals } 44 \text {. }$