MHB How to Generate a Function for Vertex Distance in a Planar Tree?

[Howang]

Hi,

Please I need you help to solve this problem:

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Consider a planar tree with $n$ non-root vertices (root edge selected).

1. Give a generating function for vertices distance $d$ from the root.
2. Proof that the total number is $$\displaystyle \binom{2n}{n-d}\frac{2d+1}{(n+d+1)}$$

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We are supposed to have an exponential generating function then use Lagrange Inversion Theorem.

 

[DreamWeaver]

Hello there, Howang! :D

Have you managed a partial proof, or made any inroads into solving this problem? If so, please DO share.

Thanks!

Gethin