A Note on Counting Dependency Trees

TL;DR

This paper develops a symbolic method to derive the generating function for counting dependency trees, uses Lagrange inversion for explicit formulas, applies Stirling's approximation for estimates, and discusses additive parameters.

Contribution

It introduces a symbolic approach combined with classical analysis techniques to analyze the counting sequence and parameters of dependency trees.

Findings

Derived a functional equation for the generating function of dependency trees

Obtained explicit formulas for counting sequences using Lagrange inversion

Provided approximations of the counting sequence via Stirling's formula

Abstract

We apply symbolic method to deduce functional equation which generating function of counting sequence of dependency trees must satisfy. Then we use Lagrange inversion theorem to obtain concrete expression of the counting sequence. We apply the famous Stirling's approximation to get approximation of the counting sequence. At last, we discuss the additive parameters of dependency trees.