Counting ternary trees according to the number of middle edges and factorizing into $(3/2)$-ary trees

TL;DR

This paper explores counting ternary trees based on middle edges, introduces a substitution to factor the generating cubic equation, and extends the concept of (3/2)-ary trees originally proposed by Knuth.

Contribution

It presents a novel method to count ternary trees by middle edges and extends the concept of (3/2)-ary trees through a new factorization approach.

Findings

Derived a new counting sequence for ternary trees

Factored the cubic generating equation using a substitution

Extended the concept of (3/2)-ary trees

Abstract

The sequence A120986 in the Encyclopedia of Integer Sequences counts ternary trees according to the number of nodes and the number of middle edges. Using a certain substition, the underlying cubic equation can be factored. This leads to an extension of the concept of $(3/2)$-ary trees, introduced by Knuth in his christmas lecture from 2014.