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

In summary, a Generating Function Tree is a graphical representation of a generating function, used in combinatorics and analysis. It is constructed by recursively applying rules to the original function and is useful for solving problems involving generating functions. It has applications in mathematics, computer science, physics, and engineering, but may have limitations with non-linear functions and multiple variables.
  • #1
Howang
1
0
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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  • #2
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
 

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

What is a Generating Function Tree?

A Generating Function Tree is a graphical representation of a generating function, which is a mathematical tool used in combinatorics and analysis to describe a sequence of numbers. It is a tree structure that shows how a sequence can be broken down into smaller subsequences and their respective generating functions.

How is a Generating Function Tree constructed?

A Generating Function Tree is constructed by recursively applying a set of rules to the original generating function until it is broken down into simpler functions. Each level of the tree corresponds to a different degree of the generating function, with the root being the original function and the leaves being the simplest functions.

What is the purpose of a Generating Function Tree?

The purpose of a Generating Function Tree is to provide a systematic way of solving problems involving generating functions. It allows for the identification of patterns and simplification of complex functions, making it easier to calculate coefficients and analyze the behavior of a sequence.

What are some applications of Generating Function Trees?

Generating Function Trees have many applications in mathematics and computer science, including counting problems, probability calculations, and algorithm analysis. They are also used in physics and engineering to solve problems involving recurrence relations and differential equations.

What are some limitations of Generating Function Trees?

Generating Function Trees are limited in their ability to solve problems involving non-linear generating functions or those with multiple variables. They also require a good understanding of generating functions and their properties to be used effectively. Additionally, they may become complex and difficult to interpret for large or highly nested generating functions.

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