# 1. Introduction

A *B+ tree* is a type of self-balancing tree structure that maintains sorted data blocks in a manner that allows binary searches, sequential access, insertions, and deletions in logarithmic amortized time. Unlike *B-trees*, in a B+ tree, keys and records are only found in the leaf nodes, leading to more efficient equality searches.

# 2. Program Steps

1. Define a *BPlusNode* class which will be the base for *InternalNode* and *LeafNode* subclasses.

2. Implement the *BPlusTree* class to handle operations like insertion, searching, and traversal.

3. Implement insertion with node splitting when nodes are overfull.

# 3. Code Program

```
class BPlusTree {
private final int T;
private BPlusNode root;
abstract class BPlusNode {
int numKeys;
int[] keys;
}
class InternalNode extends BPlusNode {
BPlusNode[] children;
InternalNode() {
keys = new int[T - 1];
children = new BPlusNode[T];
}
}
class LeafNode extends BPlusNode {
int[] values;
LeafNode nextLeaf;
LeafNode() {
keys = new int[T];
values = new int[T];
nextLeaf = null;
}
}
public BPlusTree(int t) {
T = t;
root = new LeafNode();
}
public Integer search(int key) {
return search(root, key);
}
private Integer search(BPlusNode node, int key) {
if (node instanceof LeafNode) {
for (int i = 0; i < node.numKeys; i++) {
if (node.keys[i] == key) {
return ((LeafNode) node).values[i];
}
}
return null;
} else {
for (int i = 0; i < node.numKeys; i++) {
if (key < node.keys[i]) {
return search(((InternalNode) node).children[i], key);
}
}
return search(((InternalNode) node).children[node.numKeys], key);
}
}
public void insert(int key, int value) {
// Implementation left for brevity. Involves navigating to the appropriate leaf node and inserting, possibly splitting nodes on the way back up.
}
// ... Additional methods like delete, traverse, etc. would go here ...
public static void main(String[] args) {
BPlusTree tree = new BPlusTree(3);
// Simulated insertions and searches would go here. Note: insert method not implemented in this snippet for brevity.
System.out.println("Search for key 10: " + tree.search(10));
}
}
```

### Output:

Search for key 10: null

# 4. Step By Step Explanation

1. The *BPlusTree* is defined with nodes that can be either *InternalNode* or *LeafNode* classes.

2. *InternalNode* represents intermediate nodes of the tree containing keys and children.

3. *LeafNode* represents the leaves where actual values are stored. In addition to keys and values, each leaf has a *nextLeaf* pointer, facilitating efficient range queries.

4. The *search* method allows us to look for a particular key in the tree.

5. The *insert* method, though not fully implemented here, would handle the logic of inserting keys into the tree, potentially splitting nodes if they become overfull.

Note: This is a very basic implementation of a B+ tree. A full-fledged implementation would involve much more complexity, especially in node splitting and merging logic during insertions and deletions.