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

AI Thread Summary
To generate a function for vertex distance in a planar tree, the focus is on finding an exponential generating function for the distances from the root. The discussion emphasizes using the Lagrange Inversion Theorem to derive the necessary results. A specific formula for the total number of vertices at distance $d$ from the root is presented, which is $$\displaystyle \binom{2n}{n-d}\frac{2d+1}{(n+d+1)}$$. Participants are encouraged to share any progress or partial proofs related to the problem. The conversation highlights collaborative problem-solving in combinatorial mathematics.
Howang
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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.
 
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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
 
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