### A Note on Naturally Embedded Ternary Trees

#### Abstract

In this note we consider ternary trees naturally embedded in the plane in a deterministic way. The root has position zero, or in other words label zero, and the three children of a node with position $j\in\mathbb{Z}$ have positions $j-1$, $j$, and $j+1$. We derive the generating function of embedded ternary trees where all internal nodes have labels less than or equal to $j$, with $j\in\mathbb{N}$. Furthermore, we study the generating function of the number of ternary trees of size $n$ with a given number of internal nodes with label $j$. Moreover, we discuss generalizations of this counting problem to several labels at the same time. We also study a refinement of the depth of the external node of rank $s$, with $0\le s\le 2n$, by keeping track of the left, center, and right steps on the unique path from the root to the external node. The $2n+1$ external nodes of a ternary tree are ranked from the left to the right according to an inorder traversal of the tree. Finally, we discuss generalizations of the considered enumeration problems to embedded $d$-ary trees.