Math 2.0

Combinatorics Theorem

Binary trees are counted by Catalan numbers

Builds on Catalan numbersBinary tree

Statement

Let nn be a natural number. Then the number of binary trees with exactly nn nodes is the Catalan number CnC_n.

Remarks

Here a binary tree is either empty or a root node with an ordered pair of binary trees as its left and right subtrees. For n=3n = 3 there are C3=5C_3 = 5 binary trees. Proof idea: a nonempty tree with n+1n + 1 nodes has a left subtree with ii nodes and a right subtree with n−in - i nodes. This gives the Catalan recurrence.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Binary tree, definition, no evidence levelBinary treedefinitionCatalan numbers, definition, no evidence levelCatalan numbersdefinitionBinary trees are counted by Catalan numbers, statement, no evidence levelBinary trees are coun…statement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary tree definition
  2. Catalan numbers definition
  3. Binary trees are counted by Catalan numbers statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Catalan_number, Applications in combinatorics
About this page
Tag0Y97E2DMB7
URIm2://0Y97E2DMB7
Kindstatement, theorem
WritingW0 stub
MSC05C05
StatusThis copy of the site has no build data. Run m2 status 0Y97E2DMB7.
UsesCatalan numbers, Binary tree

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