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p = 0.9;
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maxdepth = 4;
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tree[n_, f_] := (If[Length[Flatten[n]] > maxdepth, {}, 
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    If[RandomReal[] < p, 
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    If[Or[RandomReal[] < p, Length[Flatten[n]] < 2], 
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     Table[{n -> {n, i}, tree[{n, i}, f]}, {i, f}], {}]]);
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tree[k_] := 
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  Map[FromDigits[Flatten@#[[1]], k] -> 
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     FromDigits[Flatten@#[[2]], k] &, Flatten[tree[{0}, k]]];
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t = tree[3];
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GraphPlot[Flatten[t]]