How to Implement a BTree Data Structure (2023 Version)
Rudolf Bayer and Edward M. McCreight coined the term Btree data structure at the Boeing Research Labs in 1971. They published a scientific paper titled "Organization and maintenance of large ordered indices" and introduced a new data structure for fast data retrieval from disks. Although the Btree data structure has evolved over the decades, understanding its concepts is still valuable.
Here's what we'll cover in this tutorial:
 Btree data structures
 Btree properties
 Traversing in a Btree
 Searching in a Btree
 Inserting a key in a Btree
 Deleting a key in a Btree
Let's get started!
What Is a BTree Data Structure?
A Btree is a selfbalanced tree data structure that is a generalized form of the Binary Search Tree (BST). However, unlike a binary tree, each node can have more than two children. In other words, each node can have up to m children and m1 keys; also, each node must have at least $\lceil \frac{m}{2} \rceil$ children to keep the tree balanced. These features keep the tree’s height relatively small.
A Btree data structure keeps the data sorted and allows searching, inserting, and deleting operations performed in amortized logarithmic time. To be more specific, the time complexity for accomplishing the mentioned operations is $O(log {n})$, where n is the number of keys stored in the tree.
Let’s assume we have to store a tremendous amount of data in a computer system. In this common situation, we're definitely unable to store the whole data in the main memory (RAM), and most of the data should be kept on disks. We also know that the secondary storage (various types of disks) is much slower than RAM. So, when we have to retrieve the data from the secondary storage, we need optimized algorithms to reduce the number of disk access.
The BTree data structure tries to minimize the number of disk access using advanced techniques and optimized algorithms for searching, inserting, and deleting, all of which allow the Btree to stay balanced and, therefore, ensure finding data in logarithmic time.
To reveal how powerful Btree is in minimizing disk access, let’s refer to the book Introduction to Algorithms, which proves that a Btree with a height of two and 1001 children is able to store more than one billion keys. Still, only a disk access of two is required to find any key (Cormen et al., 2009).
We'll discuss the properties of Btree and how it works in the following sections.
BTree Properties
There are three types of nodes in a Btree of order m, and each of them has the following properties:
 Root Node
 A root node has between $2$ and $m$ children.
 Internal Nodes
 Each internal node has between $\lceil \frac{m}{2} \rceil$ and $m$ children — both ends are included.
 Each internal node may contain up to $m1$ keys.
 Leaf Nodes
 All leaf nodes are at the same level.
 Each leaf node stores between $\lceil \frac{m1}{2} \rceil$ and $m1$ keys — both ends are included.

The height of a BTree can be yielded via the following equation:
$h \leq log _{m}{\frac{n+1}{2}}$, where m is the minimum degree and n is the number of keys.
The illustration below shows a Btree data structure.
Traversing in a BTree
To traverse a BTree, the traversal program starts from the leftmost child and prints its keys recursively. It repeats the same process for the remaining children and keys until the rightmost child.
Searching in a BTree
Searching for a specific key in a BTree is a generalized form of searching in a Binary Search Tree (BST). The only difference is that the BST performs a binary decision, but in the Btree, the number of decisions at each node is equal to the number of the node's children.
Let's search key 102 in the Btree above:
 The root's key is not equal to 102, so since k > 100, go to the leftmost node of the right branch.

Compare k with 103; since k < 103, go to the left branch of the current node.

Compare k with the leaf's key; k exists there.
As shown, if the key we're looking for isn't in the range of the parent node, then most likely the key is in another branch. The algorithm will keep doing this to reach a leaf node. If it doesn't find the key, the algorithm returns NULL.
Inserting a Key in a BTree
So far, we've discussed traversing a Btree and searching for a specific key in a Btree data structure. This section will discuss how we can insert a new key into a BTree.
Inserting a new node into a Btree includes two steps: finding the correct node to insert the key and splitting the node if the node is full (the number of the node’s keys is greater than m1).
 If the BTree is empty:
 Allocate a root node, and insert the key.
 If the BTree is not empty:
 Find the proper node for insertion.
 If the node is not full:
 Insert the key in ascending order.
 If the node is full:
 Split the node at the median.
 Push the median key upward, and make the left keys a left child node and the right keys a right child node.
Deleting a Key in a BTree
Deletion is one of the operations we can perform on a Btree. We can perform the deletion at the leaf or internal node levels. To delete a node from a Btree, we need to follow the algorithm below:

If the node to delete is a leaf node:
 Locate the leaf node containing the desired key.
 If the node has at least $\lfloor \frac{m}{2} \rfloor$ keys, delete the key from the leaf node.
 If the node has less than $\lfloor \frac{m}{2} \rfloor$ keys, take a key from its right or left immediate sibling nodes. To do so, do the following:
 If the left sibling has at least $\lfloor \frac{m}{2} \rfloor$ keys, push up the inorder predecessor, the largest key on the left child, to its parent, and move a proper key from the parent node down to the node containing the key; then, we can delete the key from the node.
 If the right sibling has at least $\lfloor \frac{m}{2} \rfloor$ keys, push up the inorder successor, the smallest key on the right child, to its parent and move a proper key from the parent node down to the node containing the key; then, we can delete the key from the node.
 If the immediate siblings don't contain at least $\lfloor \frac{m}{2} \rfloor$ keys, create a new leaf node by joining two leaf nodes and the parent node's key.
 If the parent is left with less than $\lfloor \frac{m}{2} \rfloor$ keys, apply the above process to the parent until the tree becomes a valid BTree.

If the node to delete is an internal node:
 Replace it with its inorder successor or predecessor. Since the successor or predecessor will always be on the leaf node, the process will be similar as the node is being deleted from the leaf node.
Conclusion
This tutorial discussed Btrees and the application of Btrees for storing a large amount of data on disks. We also discussed four primary operations in Btrees: traversing, searching, inserting, and deleting. If you're interested in learning more about the actual implementation of a Btree in Python, this GitHub repo shares the implementation of the Btree data structure in Python.
I hope that you have learned something new today. Feel free to connect with me on LinkedIn or Twitter.