Editing Bernoulli number (section)

==A binary tree representation==
The Stirling polynomials {{math|''σ''<sub>''n''</sub>(''x'')}} are related to the Bernoulli numbers by {{math|''B''<sub>''n''</sub> {{=}} ''n''!''σ''<sub>''n''</sub>(1)}}. S. C. Woon described an algorithm to compute {{math|''σ''<sub>''n''</sub>(1)}} as a binary tree:{{r|Woon1997}}

:[[File:SCWoonTree.png]]

Woon's recursive algorithm (for {{math|''n'' ≥ 1}}) starts by assigning to the root node {{math|''N'' {{=}} [1,2]}}. Given a node {{math|''N'' {{=}} [''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''k''</sub>]}} of the tree, the left child of the node is {{math|''L''(''N'') {{=}} [−''a''<sub>1</sub>, ''a''<sub>2</sub> + 1, ''a''<sub>3</sub>, ..., ''a''<sub>''k''</sub>]}} and the right child {{math|''R''(''N'') {{=}} [''a''<sub>1</sub>, 2, ''a''<sub>2</sub>, ..., ''a''<sub>''k''</sub>]}}. A node {{math|''N'' {{=}} [''a''<sub>1</sub>, ''a''<sub>2</sub>, ..., ''a''<sub>''k''</sub>]}} is written as {{math|±[''a''<sub>2</sub>, ..., ''a''<sub>''k''</sub>]}} in the initial part of the tree represented above with ± denoting the sign of {{math|''a''<sub>1</sub>}}.

Given a node {{mvar|N}} the factorial of {{mvar|N}} is defined as

:<math> N! = a_1 \prod_{k=2}^{\operatorname{length}(N)} a_k!. </math>

Restricted to the nodes {{mvar|N}} of a fixed tree-level {{mvar|n}} the sum of {{math|{{sfrac|1|''N''!}}}} is {{math|''σ''<sub>''n''</sub>(1)}}, thus

:<math> B_n = \sum_\stackrel{N \text{ node of}}{\text{ tree-level } n} \frac{n!}{N!}. </math>

For example:
:{{math|1=''B''<sub>1</sub> = 1!({{sfrac|1|2!}})}}
:{{math|1=''B''<sub>2</sub> = 2!(−{{sfrac|1|3!}} + {{sfrac|1|2!2!}})}}
:{{math|1=''B''<sub>3</sub> = 3!({{sfrac|1|4!}} − {{sfrac|1|2!3!}} − {{sfrac|1|3!2!}} + {{sfrac|1|2!2!2!}})}}