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.