Let's analyze the Red-Black Tree insertion case based on the diagram: ### Tree Description: - **G** (Grandparent) - **P** (Parent, left child of G, **Red**) - **U** (Uncle, right child of G, **Black**) - **X** (New node, left child of P, **Red**) ### Red-Black Tree Insertion Cases: When a new node is inserted (X is **red**), and its parent (P) is **red**, there are three main cases to consider: 1. **Case 1 (Uncle is Red):** - Both parent (P) and uncle (U) are red. - Fix: Recolor P and U to black, recolor G to red, and repeat fix at G. 2. **Case 2 (Uncle is Black, X is "opposite" child):** - Parent (P) is red, uncle (U) is black, X is a right child of P, and P is a left child of G (or vice versa). - Fix: Perform rotation to convert to Case 3. 3. **Case 3 (Uncle is Black, X is "same side" child):** - Parent (P) is red, uncle (U) is black, X is a left child of P, and P is a left child of G (or both right). - Fix: Perform rotation and recoloring. ### Analysis of the Given Tree - **Parent (P)** is **red**. - **Uncle (U)** is **black**. - **X** is a **left child** of P, and **P** is a **left child** of G. This matches **Case 3**: - Uncle is black. - X and its parent are both on the left side (left-left). --- ## **Final Answer** **Case 3 exists.** --- ### **Summary Table** | Situation | Case | |------------------------------------------|--------------| | Uncle is red | Case 1 | | Uncle is black, X and P are opposite | Case 2 | | Uncle is black, X and P are same side | Case 3 | **Here, it's Case 3.**

Let's analyze the problem step-by-step to determine which Red-Black Tree insertion case applies, based on the diagram provided. --- ### Step 1: Understanding the Tree Structure From the image, the key nodes are: - **G**: Grandparent node - **P**: Parent node (child of G) - **U**: Uncle node (sibling of P) - **X**: New node being inserted (child of P) The colors are: - **Red** and **Black** The relationships (based on the diagram): - P is **Red** - U is **Black** - X is a **new node** being inserted as a child of P, and appears to be **Red** (since new nodes are inserted red by default) --- ### Step 2: Recall the Red-Black Tree Insertion Cases When inserting a new node, if the parent (P) is **Red**, it causes a violation of the Red-Black Tree property (no two consecutive red nodes). To fix this, three cases are considered based on the color of the uncle (U): 1. **Case 1: Uncle is Red** - Both P and U are **Red** - Fix: Recolor P and U to **Black**, recolor G to **Red**, and move up the tree to fix any violations at G. 2. **Case 2: Uncle is Black, and X is on the "opposite" side of P relative to G** - P is **Red**, U is **Black** - X is the "inner" child (e.g., right child of P when P is left child of G) - Fix: Perform a rotation at P to convert to Case 3. 3. **Case 3: Uncle is Black, and X is on the "same side" as P** - P is **Red**, U is **Black** - X is the "outer" child (left child of P when P is left child of G) - Fix: Perform rotation at G and recolor nodes to restore properties. --- ### Step 3: Analyzing the Diagram In the diagram: - P is **Red** - U is **Black** - X is inserted as **left child** of P - P is **left child** of G This configuration corresponds to **Case 3**: - Uncle U is **Black** - X is **left child** of P, and P is **left child** of G, making this a "left-left" case. --- ### **Final Answer:** **Case 3 exists.** --- ### **Explanation of Each Option:** | Option | Description | Is it applicable here? | Comments | |---------|--------------|------------------------|----------| | **Case 1** | Uncle U is **Red** | **No** | In the diagram, U is **Black**, so Case 1 does not apply. | | **Case 2** | Uncle U is **Black**, and X is an "inner" child (e.g., right child when P is left child of G) | **No** | X is a **left** child, P is a **left** child of G, so this is "left-left," not "left-right," so Case 2 does not match. | | **Case 3** | Uncle U is **Black**, and X is an "outer" child (e.g., left child when P is left child of G) | **Yes** | This matches the diagram: uncle is black, X is left child of P, P is left child of G, so this is Case 3. | --- ### **Summary:** - The configuration in the diagram corresponds to **Case 3** in Red-Black Tree insertion fixing procedures. **Therefore, the correct case is:** # **Final Answer: Case 3 exists.**

Certainly! Let's go through each of the options with detailed explanations based on the diagram and the principles of Red-Black Tree insertion cases. --- ### **Understanding the Diagram** - **Nodes:** - **G**: Grandparent node - **P**: Parent node (child of G) - **U**: Uncle node (sibling of P) - **X**: New node being inserted (child of P) - **Colors:** - **Red** and **Black** are used for nodes. - The diagram indicates **P** is **Red**, **U** is **Black**. - **X** is a **new node** inserted as **Red** (by default in insertion). - **Relationships:** - P is **left child** of G. - X is **left child** of P. - U is **right child** of G. --- ### **Recall the Red-Black Tree Insertion Cases** When inserting a new red node **X**: - If **P** (parent) is **Black**, no violation occurs. - If **P** is **Red**, we have a violation (two consecutive red nodes), requiring fixing. The fixing depends on the **uncle's color**: | Case | Uncle's Color | Description | Action | |---------|--------------|--------------|---------| | **Case 1** | **Red** | Both parent (P) and uncle (U) are **Red** | Recolor P and U to **Black**, recolor G to **Red**, and recurse at G. | | **Case 2** | **Black**, X is **inner** (e.g., right child of P when P is left child of G) | Parent is **Red**, uncle is **Black**, and X is on the **opposite side** | Rotate at P to convert to Case 3. | | **Case 3** | **Black**, X is **outer** (e.g., left child of P when P is left child of G) | Parent is **Red**, uncle is **Black**, and X is on the **same side** | Rotate at G and recolor nodes to fix properties. | --- ### **Option-by-Option Analysis** #### **Option 1: Uncle is Red** - **Description:** Uncle node U is **Red**. - **Implication:** This situation corresponds to **Case 1**. - **In the diagram:** U is **Black**; therefore, **Option 1 does not apply**. - **Conclusion:** **Not applicable here.** --- #### **Option 2: Uncle is Black, and X is on the "opposite" side (inner child)** - **Description:** Uncle U is **Black**, and **X** is the **inner** child relative to P and G. - For example: P is **left child** of G, X is **right child** of P. - **Implication:** Requires a rotation at P to convert to Case 3. - **In the diagram:** - P is **left child** of G. - X is **left child** of P. - U is **right child** of G. - Since X is **on the same side** as P (both left), this is **not** the inner child scenario. - **Conclusion:** **Not applicable here.** --- #### **Option 3: Uncle is Black, and X is on the "same side" (outer child)** - **Description:** Uncle U is **Black**, and **X** is on the **outer** side relative to P and G. - For example: P is **left child** of G, X is **left child** of P. - **In the diagram:** - P is **left child** of G. - X is **left child** of P. - U is **right child** of G. - This **matches the left-left configuration**, which is **Case 3**. - **Implication:** Perform a rotation at G and recolor nodes to fix the violation. - **Conclusion:** **This matches the current configuration, so Case 3 exists here.** --- ### **Summary of All Options** | Option | Description | Does it exist in this diagram? | Explanation | |---------|--------------|------------------------------|--------------| | **Option 1** | Uncle is **Red** | **No** | U is **Black**, so not applicable. | | **Option 2** | Uncle is **Black**, X is **inner** child | **No** | X is on the same side as P; so, not inner. | | **Option 3** | Uncle is **Black**, X is **outer** child | **Yes** | X is **left child** of P, P is **left** of G, matching Case 3. | --- ### **Final Answer:** **Case 3 exists.** --- ### **In conclusion:** - The diagram depicts a **left-left** configuration with **uncle black**, **parent red**, and **child** on the same side. - Therefore, the applicable case is **Case 3** in Red-Black Tree insertion fixing. --- **Hope this detailed explanation clarifies each option!**