A B-Tree is a self-balancing search tree that is commonly used in databases and file systems to store and retrieve large amounts of data efficiently. It has the following properties:
1. Balanced: A B-Tree is always balanced, meaning that all leaf nodes are at the same depth. This balance is achieved by ensuring that the difference in height between the left and right subtrees of any node is at most one.
2. Ordered: The keys in a B-Tree are stored in a specific order within each node. For any given node, the keys in its left subtree are smaller than the keys in the node itself, and the keys in its right subtree are larger. This ordering property allows for efficient searching and range queries.
3. Multiple Keys: Each node in a B-Tree can contain multiple keys and pointers to child nodes. The number of keys in a node is limited by the order of the B-Tree, denoted as P. For a B-Tree of order P, each node can have a minimum of ceil(P/2) keys and a maximum of P keys.
4. Flexible Structure: B-Trees are flexible in terms of their structure. They can dynamically adjust their shape and size by splitting or merging nodes as necessary to maintain balance and order. This flexibility ensures efficient insertion and deletion operations.
5. Efficient Operations: B-Trees provide efficient search, insertion, and deletion operations. The height of a B-Tree is relatively small compared to the number of keys it contains, resulting in fast access times. The complexity of these operations is typically logarithmic, making B-Trees suitable for storing and retrieving large datasets.
Now let's construct a B+ Tree with order P=4 using the given data: 10, 5, 30, 40, 20, 17, 90, 45, 93, 60, 20, 50, 29.
Step 1: Start with an empty B+ Tree.
Step 2: Insert the keys into the tree one by one, following the rules of a B+ Tree:
- Insert 10: The tree is empty, so create a new root node with 10 as the key.
```
10
```
- Insert 5: Insert it as a child of the root node.
```
10
/
5
```
- Insert 30: Insert it as a child of the root node.
```
10 30
/ /
5
```
- Insert 40: Insert it as a child of the root node.
```
10 30 40
/ / /
5
```
- Insert 20: Insert it as a child of the first child of the root node.
```
10 30 40
/ / /
5 20
```
- Insert 17: Insert it as a child of the first child of the root node.
```
10 30 40
/ / /
5 20 17
```
- Insert 90: Insert it as a child of the second child of the root node.
```
10 30 40
/ / /
5 20 17
\
90
```
- Insert 45: Insert it as a child of the second child of the root node.
```
10 30 40
/ / /
5 20 17
\
90 45
```
-
