1. Introduction

A B-tree is a self-balancing tree data structure that maintains sorted data in a way that allows searches, insertions, and deletions in logarithmic time. It is commonly used in databases and file systems to manage large amounts of data.

2. Program Steps

1. Define a BTreeNode class that contains arrays of keys and child pointers.

2. The BTree class will have methods for insertion, traversal, and search.

3. Implement insertion by splitting the nodes if they exceed maximum allowed keys.

3. Code Program

class BTree {
    private final int T;  // Minimum degree
    private BTreeNode root;

    private class BTreeNode {
        int[] keys;
        int numKeys;
        BTreeNode[] children;
        boolean isLeaf;

        BTreeNode(int t, boolean isLeaf) {
            this.isLeaf = isLeaf;
            keys = new int[2 * t - 1];
            children = new BTreeNode[2 * t];
            numKeys = 0;
        }
    }

    public BTree(int t) {
        T = t;
        root = new BTreeNode(T, true);
    }

    public void insert(int key) {
        BTreeNode r = root;
        if (r.numKeys == 2 * T - 1) {
            BTreeNode s = new BTreeNode(T, false);
            s.children[0] = r;
            splitChild(s, 0, r);
            root = s;
        }
        insertNonFull(root, key);
    }

    private void splitChild(BTreeNode x, int i, BTreeNode y) {
        BTreeNode z = new BTreeNode(T, y.isLeaf);
        x.children[i + 1] = z;
        z.numKeys = T - 1;
        System.arraycopy(y.keys, T, z.keys, 0, T - 1);
        if (!y.isLeaf) {
            System.arraycopy(y.children, T, z.children, 0, T);
        }
        y.numKeys = T - 1;
        for (int j = x.numKeys; j >= i + 1; j--) {
            x.children[j + 1] = x.children[j];
        }
        x.children[i + 1] = z;
        for (int j = x.numKeys - 1; j >= i; j--) {
            x.keys[j + 1] = x.keys[j];
        }
        x.keys[i] = y.keys[T - 1];
        x.numKeys++;
    }

    private void insertNonFull(BTreeNode x, int k) {
        int i = x.numKeys - 1;
        if (x.isLeaf) {
            while (i >= 0 && k < x.keys[i]) {
                x.keys[i + 1] = x.keys[i];
                i--;
            }
            x.keys[i + 1] = k;
            x.numKeys++;
        } else {
            while (i >= 0 && k < x.keys[i]) {
                i--;
            }
            i++;
            if (x.children[i].numKeys == 2 * T - 1) {
                splitChild(x, i, x.children[i]);
                if (k > x.keys[i]) {
                    i++;
                }
            }
            insertNonFull(x.children[i], k);
        }
    }

    public void traverse() {
        traverse(root);
        System.out.println();
    }

    private void traverse(BTreeNode x) {
        int i;
        for (i = 0; i < x.numKeys; i++) {
            if (!x.isLeaf) {
                traverse(x.children[i]);
            }
            System.out.print(x.keys[i] + " ");
        }
        if (!x.isLeaf) {
            traverse(x.children[i]);
        }
    }

    public boolean search(int k) {
        return search(root, k) != null;
    }

    private BTreeNode search(BTreeNode x, int k) {
        int i = 0;
        while (i < x.numKeys && k > x.keys[i]) {
            i++;
        }
        if (i < x.numKeys && k == x.keys[i]) {
            return x;
        } else if (x.isLeaf) {
            return null;
        } else {
            return search(x.children[i], k);
        }
    }

    public static void main(String[] args) {
        BTree tree = new BTree(3);
        tree.insert(10);
        tree.insert(20);
        tree.insert(5);
        tree.insert(6);
        tree.insert(12);
        tree.insert(30);
        tree.insert(7);
        tree.insert(17);

        System.out.println("Traverse the tree:");
        tree.traverse();

        System.out.println("Search for values:");
        System.out.println("17: " + tree.search(17));  // true
        System.out.println("100: " + tree.search(100)); // false
    }
}

Output:

Traverse the tree:
5 6 7 10 12 17 20 30
Search for values:
17: true
100: false

4. Step By Step Explanation

1. The BTree class maintains the B-Tree structure. The BTreeNode is an inner class representing each node in the B-Tree.

2. The insert method is responsible for adding a key to the tree. If the root is full, it gets split.

3. The splitChild method is used during the insertion process when a node is full.

4. The insertNonFull method recursively finds the right position for the new key in a node that is not full.

5. The traverse method is a simple in-order traversal.

6. The search method looks for a key in the tree.

7. In the main function, we insert several values into the B-Tree and then traverse it and perform some searches.

Note: This is a basic implementation of a B-tree and might not cover all the edge cases or optimizations for real-world applications.