Consider the game tree shown below. Assuming both players act optimally, use alpha-beta pruning to find the value of the root node. The search goes from left to right; when choosing which child to visit first, choose the left-most unvisited child. In the first set of boxes below, enter the values of the labeled nodes. Then, select the leaf nodes that don't get visited due to pruning.
Hint: Note that the value of a node where pruning occurs is not necessarily the maximum or minimum (depending on which node) of its children. When you prune on conditions V>β
V>β or V<α
V<α, assume that the value of the node is V
V.
 game tree: D's children are A, B, and C. A's children are 8,4, and 9. B's children are 3, 5, and 6. C's children are 7, 2, and 1. D is maximizing. A, B, and C are minimizing.

What are the values of nodes A, B, C, and D?
What leaf values that don't get visited due to pruning?

Let's solve this using the alpha-beta pruning algorithm step by step, where the search is performed from left to right. Player D is maximizing, and its children A, B, and C are minimizing nodes.

### Step-by-Step Alpha-Beta Pruning Process

**Node A: Minimizing**  
- Children: 8, 4, 9
- We begin by visiting the left-most child of A, which is 8.
  - The current minimum at A is 8.
- We then visit the second child of A, which is 4.
  - The minimum becomes 4 (since 4 < 8).
- We then visit the third child of A, which is 9.
  - The minimum remains 4 (since 4 < 9).
- **Value of A = 4.**

**Node B: Minimizing**  
- Children: 3, 5, 6
- We start by visiting the left-most child of B, which is 3.
  - The current minimum at B is 3.
- Now we apply alpha-beta pruning. Since D is maximizing and already has an α-value of 4 (from node A), we check if further exploration of node B's children is necessary.
- Since 3 < 4, player D would not choose node B anyway, so we **prune** the rest of node B's children (5 and 6).
- **Value of B = 3** (even though we don't visit all its children due to pruning).

**Node C: Minimizing**  
- Children: 7, 2, 1
- We begin by visiting the left-most child of C, which is 7.
  - The current minimum at C is 7.
- We then visit the second child of C, which is 2.
  - The minimum becomes 2 (since 2 < 7).
- Now we apply alpha-beta pruning. Since D is maximizing and already has an α-value of 4 (from node A), we check if further exploration of node C's children is necessary.
- Since 2 < 4, player D would not choose node C anyway, so we **prune** the rest of node C's children (the third child, 1).
- **Value of C = 2.**

**Node D: Maximizing**  
- Now we calculate the value of the root node D, which takes the maximum of A, B, and C.
- Values of A, B, and C are 4, 3, and 2, respectively.
- **Value of D = 4** (since 4 is the maximum of 4, 3, and 2).

### Final Values of the Nodes
- A = 4
- B = 3
- C = 2
- D = 4

### Leaf Nodes Not Visited Due to Pruning
- B's children 5 and 6
- C's child 1

These nodes are not visited because their exploration is unnecessary once pruning conditions are met.

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Do you have any questions or want more details about this process? 

Here are 5 follow-up questions you might explore:
1. What would happen if the values of B's children were different?
2. How does alpha-beta pruning reduce the complexity of the tree search?
3. How would the process change if the tree were searched right to left?
4. What would be the result if D was a minimizing node instead of maximizing?
5. How does the choice of search order (left-to-right or right-to-left) impact pruning?

**Tip**: Alpha-beta pruning can significantly speed up the decision-making process in games by eliminating branches that don't need to be explored!

Consider the game tree shown below. Assuming both players act optimally, use alpha-beta pruning to find the value of the root node. The search goes from left to right; when choosing which child to visit first, choose the left-most unvisited child. In the first set of boxes below, enter the values of the labeled nodes. Then, select the leaf nodes that don't get visited due to pruning. Hint: Note that the value of a node where pruning occurs is not necessarily the maximum or minimum (depending on which node) of its children. When you prune on conditions V > β or V < α, assume that the value of the node is V. Game tree: D's children are A, B, and C. A's children are 8, 4, and 9. B's children are 3, 5, and 6. C's children are 7, 2, and 1. D is maximizing. A, B, and C are minimizing. What are the values of nodes A, B, C, and D? What leaf values don't get visited due to pruning?

Understand how to apply alpha-beta pruning on a game tree, finding the value of nodes in a maximizing and minimizing setup. This problem includes step-by-step pruning of nodes A, B, and C and determining the root node value, D. Suitable for college-level computer science students.

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