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Binary Tree Nodes Calculator

Work out binary tree nodes instantly with clear inputs, formula shown and shareable results.

About Binary Tree Nodes Calculator

A binary tree of height h holds at most 2^(h+1) − 1 nodes when perfect and as few as h + 1 when it degenerates into a chain. Conversely n nodes need a height of at least ⌈log₂(n + 1)⌉ − 1.

Formulas

Maximum nodes

N_max = 2^(h+1) − 1

Minimum height

h_min = ⌈log₂(n + 1)⌉ − 1

Worked example

With the default values below, Binary Tree Nodes Calculator returns Maximum nodes at that height of 127. Change any input above and every figure here is recalculated in your browser.

Inputs used

Height (root is level 0)accepts 0–50
6
Actual node count to compareminimum 1
100

Results

Maximum nodes at that height
127
Minimum nodes at that height
7
Leaves when perfect
64
Minimum height for your node count
6

Step by step

  1. 1Values used
  2. 2Maximum nodes
  3. 3Minimum height
  4. 4Maximum nodes at that height= 127
  5. 5Minimum nodes at that height= 7
  6. 6Leaves when perfect= 64
  7. 7Minimum height for your node count= 6

Assumptions and limits

  • The result is derived only from the 2 inputs listed above. Anything not among them is not part of the calculation.
  • In particular, this calculator does not take inflation, tax, fees or charges or compounding frequency into account.
  • Figures are rounded for display to whole numbers, so totals built from several rounded parts can differ slightly from the unrounded arithmetic.

How these calculators are built and tested is described on the Methodology page. If a figure looks wrong, please report it.

Frequently Asked Questions

How many nodes fit in height 6?

At most 2⁷ − 1 = 127, with 64 of them leaves.

What height do 100 nodes need?

At least ⌈log₂ 101⌉ − 1 = 6, which is why balanced trees keep operations logarithmic.

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