CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-26

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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$∣ T_{n} ∣ = C_{n}$

Statement

For every natural number $n$ , the set $T_{n}$ of binary trees of size $n$ is finite and has cardinality

$∣ T_{n} ∣ = C_{n},$

the $n$ th Catalan number.

Facts & Assumptions

Given: a natural number $n$ .

[L1]

There is a bijection $T_{n} \to D_{n}$ (There is a bijection $T_{n} \to D_{n}$ for every $n$ ).

[F1]

If $A$ is finite and $f : A \to B$ is a bijection, then $B$ is finite and $∣ A ∣ = ∣ B ∣$ (The cardinality $∣ A ∣$ of a finite set).

[L2]

$C_{n} = ∣ D_{n} ∣$ (The Catalan number $C_{n} : = ∣ D_{n} ∣$ ).

Proof

technique · direct

1.1

The bijection of [L1] identifies $T_{n}$ with $D_{n}$ .

2.1

Since $D_{n}$ has cardinality $C_{n}$ by [L2], [F1] transports that cardinality along the bijection of step 1.1 and gives $∣ T_{n} ∣ = C_{n}$ .

L2F1step 1.1∎

Remarks

This is the binary-tree form of the Catalan count. Later examples use it in the forward direction, by listing trees of a fixed size, and in the backward direction, by importing a Catalan identity into the tree family.

Depends on

Used by

lvertPₙ₊₂|=Cₙ Corollary

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

D. Guichard, An Introduction to Combinatorics and Graph Theory, §3.5 (standard reference, not scraped)

∣Tn∣=Cn∣Tn​∣=Cn​