  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 $j$ have positions $j-1$, $j$,
and $j+1$, for all $j\in\Z$. We derive the generating function of ternary trees
where all nodes have labels which are less or equal than $j$, with $j\in\N$,
and the generating function of ternary trees counted with respect to nodes with
label $j$, with $j\in\Z$. 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 $s$, or in other words leaf
$s$ with $0\le s\le 2n$, where the $2n+1$ 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, right and center -- are due to the embedding of
the ternary tree in the plane. Finally, we discuss generalizations of the
considered enumeration problems to embedded $d$-ary trees.
