To complement my earlier questions:

I would now like to ask: how to perform a depth-first in-order traversal?

Wikipedia gives this illustration of the depth-first in-order traversal:

enter image description here

The output is A, B, C, D, E, F, G, H, I.

I believe this traversal applies only to binary trees.

Here is an expression to experiment with, using Null for missing leaves.

tree = "F"["B"["A", "D"["C", "E"]], "G"[Null, "I"["H", Null]]];

I am interested in efficiency and elegance.

  • 1
    $\begingroup$ Can you define in-order traversal for a generic tree (not necessarily binary)? What would that mean? $\endgroup$ Commented Mar 21, 2017 at 19:03
  • $\begingroup$ @LeonidShifrin Good question. I cannot. I found the example in isolation interesting and wondered if there were already Mathematica functions that implement this, or other standard methods. The Wikipedia description applies to a binary tree, and I don't think this generalizes to an arbitrary number of leaves, as it visits the root node between left and right branches. $\endgroup$
    – Mr.Wizard
    Commented Mar 21, 2017 at 19:34
  • 2
    $\begingroup$ For the GIH branch, how does one decide which is left and which is right? $\endgroup$
    – wxffles
    Commented Mar 21, 2017 at 20:24
  • $\begingroup$ @wxffles It was a quick fix so I self-answered instead of deleting. $\endgroup$
    – Mr.Wizard
    Commented Mar 22, 2017 at 0:03
  • $\begingroup$ @Mr.Wizard, as left and right branches have been assigned an order, why can't 3 or more branches be similarly ordered? $\endgroup$ Commented Apr 11, 2018 at 17:04

1 Answer 1



tree = "F"["B"["A", "D"["C", "E"]], "G"[Null, "I"["H", Null]]];

dfio[f_][Null | a_~r_~b_] := Scan[dfio[f], {a, r, b}]
dfio[f_][x_] := f[x]












Daniel showed how to manually manage a stack. Applying his method to this traversal:

dfioStack[f_, expr_] :=
  Module[ {stack = {expr, {}}, el = expr},
    While[ stack =!= {},
      {el, stack} = stack;
      If[ Length@el === 2,
        Do[ stack = {el[[j]], stack}, {j, {2, 0, 1}}],
        If[ el =!= Null, f @ el]

dfioStack[Print, tree]  (* same output as above *)

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