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I have a function that generates nested lists of the following form:

l1 = {{a1,a2},{{a11,a12},{a21,a22}},{{{a111,a112},{a121,a122}},{{a211,a212},{a221,a222}}},...}
l2 = {{b1,b2,b3},{{b11,b12,b13},{b21,b22,b23},{b31,b32,b33}},...}

So the nested lists contain (d (-1 + d^n))/(-1 + d) elements (where {n,d} = Dimensions@list). If they are flattened, we clearly end up with

l1Flat = {a1, a2, a11, a12, a21, a22, a111, a112, a121, a122, a211, a212, a221, a222, ...}
l2Flat = {b1, b2, b3, b11, b12, b13, b21, b22, b23, b31, b32, b33, ...}

I would like to easily reobtain the original nested list. I am sure it is easy and I am missing something here...

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If I understand your data structure, such an array might be generated as

d = 2
n = 3
a = RandomInteger[10, ConstantArray[d, #]] & /@ Range[n]
aflat = Flatten[a]

It could then be "unflatted" as

aparts = Internal`PartitionRagged[aflat, d^# & /@ Range[n]]
MapIndexed[Nest[Function[lst, Partition[lst, d]], #1, First@#2 - 1] &, aparts]
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    $\begingroup$ Recent versions of Mathematica are shipped with TakeList with the same functionality as Internal`PartitionRagged. Based on my experience, Internal`PartitionRagged is often faster than TakeList. $\endgroup$ – Henrik Schumacher Dec 6 '18 at 21:47

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