In[1]:= Level[1/2,{-1}]
Out[1]= {1/2}


enter image description here

In[2]:= Level[1/2,{-1}]
Out[2]= {1/2}

In[3]:= Level[Unevaluated[1/2],{-1}]
Out[3]= {1,2,-1}

In[4]:= 1/2//FullForm
Out[5]//FullForm= Rational[1,2]

Rational[1,2] in TreeForm,but Times[1,Power[2,-1]] when Level.

In[5]:= Level[Unevaluated[1/2],{-1},Heads->True]
Out[5]= {Times,1,Power,2,-1}

How to comprehend this? How to obtain TreeForm[Hold@Unevaluated[1/2]] without Hold and Uevaluated in whole TreeForm's graph or TreeForm@{{{Hold[1/2],b}},b,c} with out hold? enter image description here

And How to get {1,2} from Level[1/2, {-1}]?


1 Answer 1


There is an evaluation leak in TreeForm that requires a double-Unevaluated to circumvent:

TreeForm[Unevaluated @ Unevaluated[1/2]]

enter image description here

The second question is more troublesome. Because Rational is an atomic object Level does not extract its conceptual sub-parts. This is true of other atomic objects as well:

sa = SparseArray @ Range @ 5;
Level[sa, {-1}]

The only thing I can think of is a conversion to held FullForm as follows:

Level[MakeExpression @ ToBoxes @ FullForm[1/2], {-1}]
{1, 2}
Level[MakeExpression @ ToBoxes @ FullForm[sa], {-1}]
{Automatic, 5, 0, 1, 0, 5, 1, 2, 3, 4, 5, 1, 2, 3, 4, 5}
  • $\begingroup$ {Numerator[1/2], Denominator[1/2]} seems a more straightforward way to get the parts, unless one insists on using Level $\endgroup$
    – Aky
    May 23, 2013 at 12:19
  • $\begingroup$ @Aky I interpret the question as specifically "insisting on Level" rather than generically splitting a Rational object; I think Rational is only one example. $\endgroup$
    – Mr.Wizard
    May 23, 2013 at 12:22
  • $\begingroup$ @MrWizard That makes sense. When I wrote that, I hadn't realised that your technique could be used with other atomic expressions too. Thanks for clarifying. $\endgroup$
    – Aky
    May 23, 2013 at 12:27
  • $\begingroup$ @MrWizard Is there any advantage to using ToBoxes vs. ToString? (I don't actually know anything about "boxes" yet, but just asking.) $\endgroup$
    – Aky
    May 23, 2013 at 12:28
  • $\begingroup$ @Aky I updated my answer to make that more apparent. I changed from using ToString to ToBoxes because the latter should be more robust: any expression can be represented in Box form while certain things may be changed/lost in the string conversion. $\endgroup$
    – Mr.Wizard
    May 23, 2013 at 12:30

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