Your Question:

Consider a state space that contains the following two sequences of states:

(a, b, c, d, e)

Oc. 9 in a tree, 7 in a DAG

Od. None of the other answers

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O b. 125

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Oe 155

O a 7

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Oe 1

O c. 374 nodes

Od. 55 nodes

Step By Step Answers with Explanation

Question 1: How many nodes does the smallest tree containing both sequences have? And how many nodes does the smallest DAG containing both sequences have?

For the sequences (a, b, c, d, e) and (a, f, g, d, e), we need to find the smallest tree and DAG that can contain both sequences.

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c g

In this tree, there are 9 nodes.

A Directed Acyclic Graph (DAG) can have multiple roots and may have shared nodes. It should be a directed structure, and we can represent it as follows:

c g

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To find the total number of nodes in a tree with branching factor b=5 and depth d=3, we can use the formula for the total nodes in a tree:

Total Nodes = (b^(d+1) - 1) / (b - 1)

Total Nodes = 156

So, the answer is: 156 nodes.

Level 3: 12 nodes (3^2)

Level 4: 24 nodes (3^3)

In this DAG, we have a depth of d=4. Each node except the leaves has 7 children, and each node except the root and its children has 7 incoming edges.

Since the root's children have only one incoming edge, they form the first level of the DAG, and there are 7 of them. At each subsequent level, the number of nodes with 7 children increases.

Now, to find the total number of leaves, we need to sum the nodes at the last level (level 4):

Total Leaves = 2401 leaves

Level 1: 11 nodes

Level 2: 11^2 = 121 nodes

Total Nodes = 11 + 121 + 1331 + 14641 + 161051

Total Nodes = 176155 nodes