This question is similar to this: Nested list to graph. How to "flatten" an arbitrary expression, eg

expr = a[b[c, d[e][f], g], h]

to a list of key-value pairs representing Graph edges of the expression tree:

enter image description here

These can be extracted by applying WReach's exception-based method --> Can TreeForm be displayed “sideways”?:

  t_TreePlot := Throw@Hold@t;
  ][[1, 1]]


{{"a", "0", "a[b[c, d[e][f], g], h]"} -> {"b", "1", 
   "b[c, d[e][f], g]"}, {"b", "1", "b[c, d[e][f], g]"} -> {"c", "2", 
   "c"}, {"b", "1", "b[c, d[e][f], g]"} -> {"d[e]", "3", 
   "d[e][f]"}, {"d[e]", "3", "d[e][f]"} -> {"f", "4", "f"}, {"b", "1",
    "b[c, d[e][f], g]"} -> {"g", "5", "g"}, {"a", "0", 
   "a[b[c, d[e][f], g], h]"} -> {"h", "6", "h"}}

(Why are they cast to String?) Note non-atomic sub-expressions replaced with their Head. Based on the above, and the rule: ({h_, _, _} -> {a_, _, _}) :> ToExpression@h -> ToExpression@ a gives:

{a -> b, b -> c, b -> d[e], d[e] -> f, b -> g, a -> h}

TreeForm is a wrapper around TreePlot. TreePlot[%, VertexLabeling -> True] gives:

enter image description here

Since the layout is different, TreePlot must be making use of the other components of the output of the Block above.


How do TreeForm and SparseAray`ExpressionToTree (see below) extract these pairs of vertices? "Proof of work" is to extract the position (in the expression) of each vertex along with the edges.

Previously, I asked how to extract these "edges" based on a more restricted example Alternatives ordering affects pattern matching in Cases?. Also tried ReplaceList but don't know how to map it consistently across all levels.


4 Answers 4


Although kguler posted an answer using a nice internal function that does this (almost) directly I find this kind of expression manipulation interesting in itself so I wanted to see what could be done without it.

expr = a[b[c, d[e][f], g], h];

edges =
    Cases[expr, h_[___, c_[___] | c_?AtomQ, ___] /; Sow[h -> c], {0, -1}] 
  ][[2, 1]];

TreePlot[edges, VertexLabeling -> True]

enter image description here

Or for the different layout:

TreePlot[edges, Automatic, Head @ expr, VertexLabeling -> True]

enter image description here

A user asked about non-unique node names. Here is a first attempt at addressing that case.

tree[expr_] :=
  Module[{e2, edges, head},
    e2 = MapIndexed[head, expr, {0, -1}, Heads -> True];
    edges = Reap[
       Cases[e2, (head[h_, x_])[___, 
          head[head[c_, {z__}][___] | c_?AtomQ, {a__}], ___] /; 
         Sow[Annotation[h, x] -> 
           Annotation[c, If[{z} === {}, {a}, {z}]]], {0, -1}]][[2, 1]];
    edges = edges /. head -> Annotation;
    TreePlot[edges, Automatic, edges[[-1, 1]], VertexLabeling -> True]

a[b[c, d[e][f], g, b, d[e][b]], h] // tree

enter image description here

This very likely has bugs that will need to be addressed as I did it in a hurry and tired, but I think it at least gives us a place to start.

  • $\begingroup$ Thanks, nice use of Sow/Reap, and inner Alternatives. One issue, how to specify Head[expr] as RootVertex? Maybe using a function of Position[expr, h] (in Sow)? $\endgroup$ Jun 27, 2014 at 21:09
  • $\begingroup$ @alancalvitti I'm glad this helps. Regarding "RootVertex" does this do what you want? TreePlot[%, Automatic, Head@expr, VertexLabeling -> True] $\endgroup$
    – Mr.Wizard
    Jun 28, 2014 at 1:11
  • $\begingroup$ this answer, as the others here, relies on the nodes being all different. it's not easy to see how to extend it to the case where e.g. b appears twice at different places. one would probably need some temporary node numbering? $\endgroup$ Sep 12, 2017 at 18:14
  • $\begingroup$ @user3240588 Good question! I'll have to think about that. $\endgroup$
    – Mr.Wizard
    Sep 12, 2017 at 20:47
  • $\begingroup$ @user3240588 Please see my update, and look for cases where it fails so I can try to fix those. $\endgroup$
    – Mr.Wizard
    Sep 12, 2017 at 21:25

An alternative method to WReach's method is to use SparseArray`ExpressionToTree which produces the same output without string wrappers:

expr = a[b[c, d[e][f], g], h];
ett = SparseArray`ExpressionToTree[expr]
(* {{a,0,a[b[c,d[e][f],g],h]}->{b,1,b[c,d[e][f],g]},
    {a,0,a[b[c,d[e][f],g],h]}->{h,6,h}} *)

edges = ett[[All,All,1]] (* thanks: @Mr.Wizard *)
(* or edges = ett /. Rule[a_, b_] :> Rule[First[a], First[b]];*)
(* {a->b,b->c,b->d[e],d[e]->f,b->g,a->h} *)

Graph[edges, VertexLabels -> Placed["Name", {Center, Center}],
             VertexSize -> .3, VertexLabelStyle -> Directive[Red, Italic, 20],
             ImagePadding -> 20, ImageSize -> 400, 
             GraphLayout -> {"LayeredEmbedding", "RootVertex" -> edges[[1,1]]}]

enter image description here

Update: You can also use GraphComputation`ExpressionGraph:

eg = GraphComputation`ExpressionGraph[expr, VertexSize -> Large, 
   VertexLabelStyle -> Directive[Red, Italic, 20]];
SetProperty[eg,  VertexLabels -> {v_ :> 
    Placed[PropertyValue[{eg, v}, VertexLabels], Center]}]

enter image description here

  • 3
    $\begingroup$ More directly: SparseArray`ExpressionToTree[expr][[All, All, 1]]. Also, thanks for showing me ExpressionToTree! (Again?) $\endgroup$
    – Mr.Wizard
    Jun 27, 2014 at 1:24
  • $\begingroup$ @kguler, +1 - btw ETT --> ReleaseHold. But I'm asking how TreeForm (and ETT) convert the expr. I edited my Q to reflect this. $\endgroup$ Jun 27, 2014 at 18:40
  • $\begingroup$ @alancalvitti Is my answer of no interest to you? Are you only interested in the details of the internal implementation and not an equivalent method in top-level code? $\endgroup$
    – Mr.Wizard
    Jun 27, 2014 at 19:37
  • $\begingroup$ Yes that works, thank you, I had to test on some expressions of interest. $\endgroup$ Jun 27, 2014 at 21:06

IGraph/M now includes IGExpressionTree:


enter image description here

(* True *)

Is this what you are seeking?

expr = a[b[c, d[e][f], g], h]
boxes = ToBoxes@TreeForm[expr]
positions = Cases[boxes, LineBox[{x__}] -> x, Infinity]
nodes =
    StyleBox[x_, __] :> ToExpression@x, Infinity] /. 
      {t_Times :> First@t, Verbatim[HoldForm][x_] -> x}
Rule @@@ Extract[nodes, List /@ positions]

{a -> b, a -> h, b -> c, b -> d, b -> g, d -> f}

  • $\begingroup$ No, for example that doesn't match d[e]. But also, How to extract it directly without using TreeForm? $\endgroup$ Jun 26, 2014 at 23:15

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