Enumeration of multi-rooted plane trees

TL;DR

This paper derives closed-form formulas, recursion relations, and generating functions for counting multi-rooted plane trees with specified root degrees, revealing new identities and conjectures in combinatorics.

Contribution

It introduces explicit formulas and recursion relations for enumerating multi-rooted plane trees, expanding understanding of their combinatorial properties.

Findings

Derived explicit formulas for multi-rooted plane trees.

Established recursion relations and generating functions.

Discovered a new binomial identity and hypergeometric conjecture.

Abstract

We give closed form expressions for the numbers of multi-rooted plane trees with specified degrees of root vertices. This results in an infinite number of integer sequences some of which are known to have an alternative interpretation. We also propose recursion relations for numbers of such trees as well as for the corresponding generating functions. Explicit expressions for the generating functions corresponding to plane trees having two and three roots are derived. As a by-product, we obtain a new binomial identity and a conjecture relating hypergeometric functions.