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A B+ tree stores records in linked leaves and uses high branching factor to limit block-oriented index access to logarithmically many page reads.
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Formula
n records; m maximum children per internal node; h levels; all are dimensionless counts
| Symbol | Meaning | Unit |
|---|---|---|
| Number of stored keys or records | count | |
| Maximum number of children of an internal node | count | |
| Tree height in levels | levels |
h=O(\log_{\lceil m/2\rceil} n)For order m, non-root internal nodes have at least ceil(m/2) children, all leaves are at equal depth, and records reside in leaves; internal separator keys direct the search. The height is logarithmic with branching base at least ceil(m/2). Insertions and deletions may split or merge pages but remain asymptotically logarithmic. Minimum occupancy and the definition of order vary among implementations.
For order m, non-root internal nodes have at least ceil(m/2) children, all leaves are at equal depth, and records reside in leaves; internal separator keys direct the search. The height is logarithmic with branching base at least ceil(m/2). Insertions and deletions may split or merge pages but remain asymptotically logarithmic. Minimum occupancy and the definition of order vary among implementations.
A B+ tree stores records in linked leaves and uses high branching factor to limit block-oriented index access to logarithmically many page reads.
Defining relation
The equation states the defining relation within its stated domain.
Use the example for B+ tree: height and index access to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
A B+ tree stores records in linked leaves and uses high branching factor to limit block-oriented index access to logarithmically many page reads.
The relation applies only under its stated assumptions.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: For order m, non-root internal nodes have at least ceil(m/2) children, all leaves are at equal depth, and records reside in leaves; internal separator keys direct the search. The height.
Formula cluster
A B+ tree stores records in linked leaves and uses high branching factor to limit block-oriented index access to logarithmically many page reads.
With a minimum fanout of 50 and a maximum of 100, a tree indexing 1,000,000 records needs about ceil(log base 50 of 1,000,000)=4 branching steps to reach the leaf region in this height model. Actual page height depends on root and leaf occupancy; all leaves remain at the same depth.
Understand database and file-system indexes for block-oriented range queries.
Here is how to work through a typical B+ tree: height and index access (Höhe h = O(log fanout n)) task step by step:
Task
Use the example for B+ tree: height and index access to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
A B+ tree stores records in linked leaves and uses high branching factor to limit block-oriented index access to logarithmically many page reads.
The relation applies only under its stated assumptions.
A B+ tree is a balanced multiway index: internal keys guide searches, while records reside in leaves at equal depth. Height limits page accesses, and high fanout keeps the tree shallow. With a minimum fanout of 50 and a maximum of 100, a tree indexing 1,000,000 records needs about ceil(log base 50 of 1,000,000)=4 branching steps to reach the leaf region in this height model. Actual page height depends on root and leaf occupancy; all leaves remain at the same depth. Actual height depends on root and leaf occupancy, so the asymptotic relation is not an exact height for every implementation. Validity limit: For order m, non-root internal nodes have at least ceil(m/2) children, all leaves are at equal depth, and records reside in leaves; internal separator keys direct the search. The height Source: OpenDSA, §17.6.1.1 B+ Trees and §17.6.1.2 B-Tree Analysis.
First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example With a minimum fanout of 50 and a maximum of 100, a tree indexing 1,000,000 records needs about ceil(log base 50 of 1,000,000)=4 branching steps to reach the leaf region in this height model. Actual page height depends on root and leaf occupancy; all leaves remain at the same depth. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: For order m, non-root internal nodes have at least ceil(m/2) children, all leaves are at equal depth, and records reside in leaves; internal separator keys direct the search. The height Source: OpenDSA, §17.6.1.1 B+ Trees and §17.6.1.2 B-Tree Analysis.
The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: For order m, non-root internal nodes have at least ceil(m/2) children, all leaves are at equal depth, and records reside in leaves; internal separator keys direct the search. The height Source: OpenDSA, §17.6.1.1 B+ Trees and §17.6.1.2 B-Tree Analysis.
The stated units are: n records; m maximum children per internal node; h levels; all are dimensionless counts. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: For order m, non-root internal nodes have at least ceil(m/2) children, all leaves are at equal depth, and records reside in leaves; internal separator keys direct the search. The height Source: OpenDSA, §17.6.1.1 B+ Trees and §17.6.1.2 B-Tree Analysis.
Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source OpenDSA, §17.6.1.1 B+ Trees and §17.6.1.2 B-Tree Analysis. supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: For order m, non-root internal nodes have at least ceil(m/2) children, all leaves are at equal depth, and records reside in leaves; internal separator keys direct the search. The height Source: OpenDSA, §17.6.1.1 B+ Trees and §17.6.1.2 B-Tree Analysis.
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Curated exam set for "B+ tree: height and index access":
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Which formula describes B+ tree: height and index access?
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How do you rearrange B+ tree: height and index access for Defining relation?
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Which common mistake happens with B+ tree: height and index access?
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