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 ∈ 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 2 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 ≤ s ≤ 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.