# [seqfan] Re: Help needed with A007863 Number of hybrid binary trees with n nodes.

Benoît Jubin benoit.jubin at gmail.com
Fri May 18 09:33:22 CEST 2012

```Sorry, I messed up the definition of associativity, although what to
do is quite obvious.  When you have two nodes labelled 'a', one the
child of the other, you can do the transformation which corresponds to
going from the graph representing the evaluation of (xy)z, to the one
for x(yz), the labels staying unchanged.  I hope this is clear... (the
picture is on the second page of Pallo's paper).

Benoît

On Fri, May 18, 2012 at 12:24 AM, Benoît Jubin <benoit.jubin at gmail.com> wrote:
> Indeed, the definition should be changed too:
>
> A007863 Number of hybrid binary trees with n internal nodes (hence 2n+1 nodes)
>
> and a definition of hybrid binary trees could be add in the comments
> (this is the definition used by Pallo in the reference cited there):
>
> An (a,n)-labelled binary tree is a binary tree where each internal
> node is labelled by "a" (for associative) or "n" (for nonassociative).
> We define on the set of (a,n)-labelled binary trees with m nodes an
> equivalence relation as follows: if a tree has an 'a' node A with a
> child being also an 'a' node, then if we switch the left and right
> subtrees of that node A, we obtain an equivalent tree, and all
> equivalent trees are obtained in this way.
> A hybrid binary tree is an equivalence class of (a,n)-labelled binary
> trees under this relation.
>
> Regards,
> Benoit
>
>
>
>
> On Thu, May 17, 2012 at 7:31 PM, David Scambler <dscambler at bmm.com> wrote:
>> https://oeis.org/A007863
>>
>> I am not sure what the hybrid binary trees are.
>>
>> For example, what are the 7 trees counted by a(2) = 7?
>>
>> Thanks
>> dave
>>
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```