Dangers of five-minute ex-tempore thinking....

[Antti Karttunen]

I wrote:

> Of course, for the Catalans we get also a purely combinatorial proof.
> E.g., if we use the interpretation C_n gives the number of
> "rooted plane binary trees with n+1 leaves (2n edges, 2n+1 vertices)"
> we get the following five possible planar binary trees:
>
> \/
>  \/    (and its mirror image)
>   \/
>
>   \/
>  \/    (and its mirror image)
>   \/
>
> \/ \/
>  \./
>
> and this last, symmetric tree is one which makes the result odd.
> (And similar palindromic trees may occur only when there are 2^n leaves).

Oops, here's a simple counter-example:

\/   \/
 \/ \/
  \./

so evidently, the number of such palindromic rooted plane binary
trees is even when the number of leaves is even, but not power of 2.
This is easy to see, if we take successive subtrees of such symmetric tree,
which, if themselves not symmetric, can thus be reflected (together with
their mirror image counterparts on the other side of the tree)
so that the whole tree still stays symmetric.



Yours again,

Antti Karttunen