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Data Structures/GPT Answers to Past Paper/Question 10.md

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+To insert a node into a **binary search tree (BST)** while keeping the tree **balanced**, we need to perform the following steps:
+
+1. **Insert the node** in the appropriate place according to the **BST properties** (left child < parent < right child).
+2. After insertion, **rebalance the tree** (if necessary) to ensure that it remains balanced. A tree is considered balanced if the height difference between the left and right subtrees of any node is at most 1 (this is the **balanced binary tree** or **AVL tree** property).
+
+### **Given Binary Search Tree**:
+```
+       10
+      /  \
+     5   15
+    /    / \
+   2    12  20
+```
+
+### **Step 1: Insert 4 into the BST**
+
+We start by inserting **4** into the BST:
+
+- 4 is less than 10, so we move to the left of 10.
+- 4 is less than 5, so we move to the left of 5.
+- 4 is greater than 2, so 4 will be placed as the **right child** of 2.
+
+After insertion, the tree looks like this:
+```
+       10
+      /  \
+     5   15
+    /    / \
+   2    12  20
+    \
+     4
+```
+
+### **Step 2: Balance the Tree (If Needed)**
+
+At this point, the tree is not balanced. Let's check the balance factors of the nodes:
+
+- **Node 2** has a left subtree height of 0 and a right subtree height of 1. Its balance factor is `0 - 1 = -1`, which is acceptable.
+- **Node 5** has a left subtree height of 1 and a right subtree height of 1. Its balance factor is `1 - 1 = 0`, which is balanced.
+- **Node 10** has a left subtree height of 2 (because of nodes 5, 2, and 4) and a right subtree height of 2 (because of nodes 15, 12, and 20). Its balance factor is `2 - 2 = 0`, which is balanced.
+
+Since the tree is balanced at all levels, **no further rebalancing is necessary**.
+
+### **Final Balanced Tree**:
+
+The final tree after inserting 4 is:
+```
+       10
+      /  \
+     5   15
+    /    / \
+   2    12  20
+    \
+     4
+```
+
+### **Conclusion**:
+
+After inserting 4, the tree remains balanced and no rotations are needed.