59 lines
1.9 KiB
Markdown
59 lines
1.9 KiB
Markdown
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. |