Generating function for vertices distance from the root in a planar tree

Question

I need you help to solve this problem:

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Consider a planar tree with $n$ non-root vertices.

1. Give a generating function for vertices distance $d$ from the root.

2. Proof that the total number is $$\displaystyle \binom{2n}{n-d}\frac{2d+1}{(n+d+1)}$$

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

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