CorollaryStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-26

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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Rooted plane binary trees satisfy $B (x) = 1 + x B (x)^{2}$

Statement

Let $B (x)$ be the generating function of rooted plane binary trees, specified by

$B = E + Z \times B^{2} .$

Then $B (x)$ is the unique formal power series satisfying

$B (x) = 1 + x B (x)^{2} .$

Facts & Assumptions

Given: The recursive specification $B = E + Z \times B^{2}$ .

[L1]

Disjoint union and Cartesian product translate to addition and multiplication of ordinary generating functions (Disjoint union and Cartesian product translate to addition and multiplication of ordinary generating functions).

[L2]

An order-raising recursive specification has a unique solution (An order-raising recursive specification has a unique solution).

[L3]

Formal order is non-Archimedean under sums and satisfies ${ord}_{x} (f g) \geq {ord}_{x} (f) + {ord}_{x} (g)$ over a commutative ring (Formal order is non-Archimedean under sums and additive under products over a domain).

Proof

technique · direct

1.1

The associated operator is $F (Y) = 1 + x Y^{2}$ . For any $U, V$ , one has $F (U) - F (V) = x (U + V) (U - V)$ , so [L3] gives ${ord}_{x} (F (U) - F (V)) \geq {ord}_{x} (U - V) + 1$ . Thus the specification is order-raising.

L3algebra

2.1

By [L2], the specification has a unique formal power series solution $B (x)$ .

step 1.1L2

3.1

The neutral class contributes $1$ , the atomic class contributes $x$ , and the ordered pair of left and right subtrees contributes $B (x)^{2}$ by [L1]. Hence the specification translates to $B (x) = 1 + x B (x)^{2}$ .

step 2.1L1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

14 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

Philippe Flajolet and Robert Sedgewick, Analytic Combinatorics (standard reference, not scraped)
Stephen Melczer, An Invitation to Enumeration, Chapter 5: Combinatorial Constructions (standard reference, not scraped)

Rooted plane binary trees satisfy B(x)=1+xB(x)2B(x)=1+xB(x)2