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I would like make the attached image by Mathematica. someone who could help me? Thank!enter image description here

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3  
Or CompleteKaryTree –  Öskå Jun 12 at 14:20
    
And at least but not last.., have you even tried anything? Like getting the coordinates of the points? –  Öskå Jun 12 at 15:05
    
Now I just saw the comments ... thanks everyone for the help –  user15850 Jun 13 at 6:46

2 Answers 2

Not a perfect match but a start

Partition[Table[Column[{"i = " <> ToString[2^(n - 1)], 
 CompleteKaryTree[n, 
  ImagePadding -> 
   10, (AbsoluteOptions[
     CompleteKaryTree[n, VertexLabels -> "Name", ImagePadding -> 10], 
     VertexCoordinates] /. {x_?NumericQ, y_?NumericQ} :> {x, -y})], 
 "\[Omega] = " <> ToString[n]}, Center], {n, 2, 5}], 2] // Grid

enter image description here

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If you have Version 9, you can use the GraphLayout suboptions as follows:

bbt[n_] := 
Labeled[GraphUnion[CompleteKaryTree[n], Graph[{UndirectedEdge[0, 1]}],
                   BaseStyle ->Directive[Opacity[1], Black, PointSize[.015]],
                   EdgeStyle -> Directive[{Black, Thick}], AspectRatio -> 1,
                   VertexShapeFunction -> "Point",  ImageSize -> 300,
                   GraphLayout -> {"LayeredEmbedding", "RootVertex" -> 0,
                                   "Orientation" -> Bottom, 
                                   LayerSizeFunction -> (If[# == 1, 1, 2^(n - #)] &)}],
       {"i = " <> ToString[2^(n - 1)],"\[Omega] = " <> ToString[n]}, 
       {Top, Bottom}];

Row[bbt /@ Range[2,7], Spacer[5]]

enter image description here

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I'm afraid that it doesn't look the same on v8. None of the GraphLayout options seem to work for me. –  Öskå Jun 13 at 9:26
    
@Öskå, right - these GraphLayout options are new-in-version-9. –  kguler Jun 13 at 9:31
    
Well, they seem to be very useful! You already have my +1 –  Öskå Jun 13 at 9:38
    
@Oska, thank you for the +. –  kguler Jun 13 at 9:50
    
Well, your output looks like what the OP desired, the raw one from CompleteKaryTree is getting flatter and flatter. –  Öskå Jun 13 at 9:52

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