Computer Science · Algorithms
B-Tree Order Calculator
Calculates B-tree structural properties including maximum keys, minimum keys, maximum children, tree height, and node capacity given tree order and record count.
Calculator
Formula
t is the minimum degree (order parameter): every non-root node must have at least t−1 keys and at most 2t−1 keys. Every internal non-root node has between t and 2t children. n is the total number of keys stored in the tree. h is the maximum tree height (number of levels above the leaves). The height bound shows that a B-tree of minimum degree t storing n keys has height at most log base t of (n+1)/2.
Source: Cormen, Leiserson, Rivest & Stein — Introduction to Algorithms, 4th ed., Chapter 18: B-Trees.
How it works
A B-tree of minimum degree t (also called order t or sometimes written as order 2t depending on the textbook convention) is a self-balancing search tree where every node other than the root must hold at least t − 1 keys and at most 2t − 1 keys. The root must hold at least one key (unless the tree is empty). Because all leaves sit at the same depth, the tree remains perfectly height-balanced after insertions and deletions. The minimum degree parameter directly controls the fan-out of the tree — higher fan-out means fewer levels and fewer disk reads to locate any given key.
The core formulas derive directly from the definition. Maximum keys per node = 2t − 1; minimum keys per node = t − 1; maximum children per internal node = 2t; minimum children per non-root internal node = t. The height upper bound h ≤ logt((n+1)/2) comes from counting the minimum possible number of keys in a tree of height h: at minimum, the root has 1 key, and every other node has t−1 keys, giving at least 2th−1 − 1 keys total. Solving for h yields the logarithmic bound. The minimum fill factor — the ratio of minimum keys to maximum keys per node — equals (t−1)/(2t−1), which approaches 50% as t grows, explaining why B-trees can waste up to half their allocated node space in the worst case.
In practice, database management systems such as PostgreSQL, MySQL InnoDB, SQLite, and Oracle use variants of B-trees (most commonly B+ trees, where all records are stored in leaf nodes) for table indexes and primary key structures. File systems including NTFS, HFS+, Btrfs, and XFS use B-trees or B+ trees to store directory entries and extent maps. Choosing the minimum degree is a tuning decision: t is typically chosen so that a node fills exactly one disk page or a small multiple thereof, minimising the number of page reads required per lookup. For a 4 KB disk page storing 8-byte keys and 8-byte pointers, t is typically around 100–200, giving trees that stay under 3–4 levels even for billions of records.
Worked example
Suppose you are designing a database index with minimum degree t = 50 to store n = 1,000,000 keys.
Step 1 — Node capacity: Maximum keys per node = 2(50) − 1 = 99 keys. Minimum keys per node = 50 − 1 = 49 keys. Maximum children per internal node = 2(50) = 100 children.
Step 2 — Tree height: h ≤ log50((1,000,000 + 1) / 2) = log50(500,000.5) = ln(500,000.5) / ln(50) ≈ 13.122 / 3.912 ≈ 3.35, so the maximum height is 4 levels. In practice the tree is likely only 3 levels deep with good fill.
Step 3 — Minimum fill factor: (t−1)/(2t−1) = 49/99 ≈ 49.5%. This means that in the worst case, node space utilisation falls to about half of maximum capacity.
Interpretation: With t = 50, any of the 1 million keys can be found in at most 4 disk page reads, regardless of insertion order. If each disk read takes 5 ms (spinning HDD), the worst-case lookup time is about 20 ms — a compelling argument for using high-degree B-trees in disk-backed storage engines.
Limitations & notes
This calculator implements the classic B-tree model from CLRS and may differ from the specific variant used by a given database or file system. Most real-world systems use B+ trees rather than plain B-trees: in a B+ tree, all data records are stored in leaf nodes only, and internal nodes store only keys used for routing, which increases the effective fan-out significantly. The height bound computed here is a worst-case upper bound; actual tree height for a randomly ordered insertion sequence is typically one or two levels less. The minimum degree t should be chosen to match the page size and actual key/pointer byte widths — this calculator does not perform page-size fitting. Additionally, the fill factor formula assumes worst-case (minimum occupancy) nodes throughout; in practice, most B-tree implementations maintain significantly higher average fill rates through techniques such as delayed splitting, bulk loading, and fill-factor tuning parameters exposed by the database engine.
Frequently asked questions
What is the difference between B-tree order and minimum degree?
Different textbooks define B-tree order differently. In the CLRS convention used here, the minimum degree t means every non-root node has between t−1 and 2t−1 keys. In the Knuth convention, the order m of a B-tree is the maximum number of children per node, so m = 2t. Always check which convention a reference uses before applying formulas.
Why do databases use B+ trees instead of plain B-trees?
In a B+ tree, all actual data records are stored in the leaf level, and internal nodes hold only separator keys for routing. This means internal nodes can hold more keys (higher fan-out), reducing tree height. It also makes range scans efficient because all leaves are linked in a doubly-linked list, allowing sequential traversal without backtracking through the tree.
How do I choose the optimal minimum degree t for my database index?
The optimal t is determined by your disk page size and the byte size of your key-pointer pairs. Compute t = floor((PageSize + sizeof(key)) / (2 * (sizeof(key) + sizeof(pointer)))). For a 8 KB page with 8-byte keys and 8-byte pointers, this gives t ≈ 256, meaning up to 511 keys and 512 children per node — a very high fan-out that keeps trees under 4 levels for hundreds of billions of records.
What does tree height mean in terms of disk I/O cost?
In a disk-backed B-tree, each node typically corresponds to one disk page. A lookup requires reading one page per level from root to leaf, so the tree height directly equals the number of disk reads (I/Os) per lookup. Since modern SSDs handle random reads in 50–200 μs and spinning HDDs in 5–10 ms, reducing height by even one level can cut lookup latency significantly at scale.
Can a B-tree have a minimum degree of 1?
No. A minimum degree of t = 1 would mean nodes could have zero keys and one child, which degenerates to a linked list. The B-tree definition requires t ≥ 2. With t = 2, a B-tree is sometimes called a 2-3-4 tree, where each node holds 1 to 3 keys and 2 to 4 children. These are conceptually important but impractical for disk storage due to their very low fan-out.
Last updated: 2025-01-15 · Formula verified against primary sources.