Different highlight for each cluster in DendrogramPlot

Question

Is there any way to have each cluster in the DendrogramPlot highlight in a different color? I've tried the obvious way but it doesn't work:

Needs["HierarchicalClustering`"]
data = {{0.758587,0.844749},{1.05549,1.01289},{1.21689,1.1919},{0.53086,0.977943},{2.08582,1.8978},{2.10121,1.81776},{1.88505,1.83257},{1.89067,2.48317},{2.88711,1.02682},{2.82199,0.901863},{2.90456,0.91189},{2.80032,1.09192}};
DendrogramPlot[
 data,
 LeafLabels -> Range[12],
 HighlightLevel -> 3,
 HighlightStyle -> {Red, Green, Blue}
 ]

Answer 1

If you want something like this :

(colors are random) the code is :

dendogram = 
   DendrogramPlot[data, LeafLabels -> Range[12], HighlightLevel -> 3, 
    HighlightStyle -> {Red, Green, Blue}];

 Show[
  dendogram,
  Graphics[(Cases[dendogram, Rectangle[___], {1, Infinity}] // 
      SortBy[#, -#[[2, 2]] &] & ) /. 
    x : Rectangle[___] :> {Hue[RandomReal[]], x}],
  Graphics[Cases[dendogram, Line[___], {1, Infinity}]]
  ]

or like this :

(agglomerate colors, colors not random) then :

Needs["HierarchicalClustering`"]
data = {{0.758587, 0.844749}, {1.05549, 1.01289}, {1.21689, 
    1.1919}, {0.53086, 0.977943}, {2.08582, 1.8978}, {2.10121, 
    1.81776}, {1.88505, 1.83257}, {1.89067, 2.48317}, {2.88711, 
    1.02682}, {2.82199, 0.901863}, {2.90456, 0.91189}, {2.80032, 
    1.09192}};

dendogram = 
  DendrogramPlot[data, LeafLabels -> Range[12], HighlightLevel -> 3, 
   HighlightStyle -> {Red, Green, Blue}];

intervalList = Cases[dendogram, Rectangle[___], {1, Infinity}] //
   (*Take[#,3]& //*)
   (Interval[{#[[1, 1]], #[[2, 1]]}] & /@ # &) //
  FixedPoint[
    Replace[#, {a0___, i1_, a1___, i2_, a3___} /; 
        IntervalIntersection[i1, i2] =!= Interval[] :> {a0, 
        IntervalUnion[i1, i2], a1, a3}] &, #] &

ClearAll[color];
color[1] = Red; color[2] = Green;
color[3] = Blue; color[_] = Hue[RandomReal[]];

Show[
 dendogram,
 Graphics[
  Cases[dendogram, Rectangle[___], {1, Infinity}] /. 
   x : Rectangle[___] :> {With[{pos = x[[1, 1]]}, 
      color[
       Position[intervalList, _?(IntervalMemberQ[#, pos] &)][[1, 
         1]]]], x}], 
 Graphics[Cases[dendogram, Line[___], {1, Infinity}]]
 ]

Answer 2

 Needs["HierarchicalClustering`"] 

 ClearAll[highlightLevels];
 highlightLevels[data_, level_Integer: 2, colors_List: ColorData[1, "ColorList"], opts : OptionsPattern[]] :=
 Module[{dndrgm =  DendrogramPlot[data, LeafLabels -> Automatic, HighlightLevel -> level],
 rectCoords, lines, labels, splts, newCoords, newRectangles,  colorlist},
 If[level == 1,
  DendrogramPlot[data, LeafLabels -> Automatic, HighlightLevel -> 1, 
  HighlightStyle -> colors[[1]], opts],
  rectCoords = Cases[dndrgm, Rectangle[x__] :> {x}, {0, Infinity}];
  lines = Cases[dndrgm, Line[__], {0, Infinity}];
  labels = Cases[dndrgm, Text[__], {0, Infinity}];
  splts = Split[SortBy[ rectCoords, #[[1, 1]] &], (#1[[1, 1]] <= #2[[1, 1]] <= #1[[2, 1]] &)];
 newCoords = Join @@@ Gather[splts, #2[[1, 1, 1]] <= Max@#1[[;; , 2, 1]] &];
 newRectangles = Rectangle @@@ # & /@ newCoords;
 colorlist = colors[[Mod[#, Length[colors], 1]]] & /@  Range[Length[newRectangles]]; 
 Graphics[
Join[Riffle[colorlist, newRectangles], {Black}, lines, labels], PlotRange -> All, opts]]]

Example:

data = {{0.758587, 0.844749}, {1.05549, 1.01289}, {1.21689, 1.1919},
 {0.53086, 0.977943}, {2.08582, 1.8978}, {2.10121, 1.81776}, 
 {1.88505, 1.83257}, {1.89067, 2.48317}, {2.88711, 1.02682},
 {2.82199, 0.901863}, {2.90456, 0.91189}, {2.80032, 1.09192}};
Grid[Partition[highlightLevels[data, #, ColorData[1, "ColorList"][[2 ;;]],
 AspectRatio -> 1, ImageSize -> 250] & /@ Range[9], 3],  Dividers -> All]