A note on naturally embedded ternary trees
Markus Kuba

TL;DR
This paper studies the enumeration and properties of naturally embedded ternary trees with planar structure, deriving generating functions for various label constraints and analyzing node depths, with extensions to d-ary trees.
Contribution
It introduces generating functions for labeled ternary trees with planar embedding and explores depth distributions, extending to general embedded d-ary trees.
Findings
Derived generating functions for labeled ternary trees.
Analyzed depths of external nodes in embedded trees.
Extended enumeration methods to d-ary trees.
Abstract
In this note we consider ternary trees naturally embedded in the plane in a deterministic way such that the root has position zero, or in other words label zero, and the children of a node with position have positions , , and , for all . We derive the generating function of ternary trees where all nodes have labels which are less or equal than , with , and the generating function of ternary trees counted with respect to nodes with label , with . Moreover, we discuss generalizations of the counting problem to several labels at the same time. Furthermore, we use generating functions to study the depths of the external node , or in other words leaf with , where the external nodes of a ternary tree are numbered from the left to the right according to an inorder traveral. The three different types depths -- left,…
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Taxonomy
TopicsAdvanced Combinatorial Mathematics · Graph theory and applications · Graph Labeling and Dimension Problems
