list manipulation - How to enumerate all possible binary associations?

Suppose I have a list of symbols like:

{a,b,c,d}

I would like to enumerate all possible binary associations (combining symbols and/or sublists pairwise):

{{{a,b},c},d}
{{a,b},{c,d}}

{a,{b,{c,d}}}
{{a,{b,c}},d}
{a,{{b,c},d}}

There should be altogether 5 solutions for this example. My question is how can I enumerate all such associations for a generic list?

I have tried

ReplaceList[{a,b,c,d},{u___,v_,w_,x___}:>{u,{v,w},x}]

But this only works for the first layer.

Answer

I propose a more compact approach

f[list__] := Join @@ ReplaceList[{list}, {x__, y__} :> Tuples@{f[x], f[y]}]
f[x_] := {x};

f[a, b, c, d] // Column

{a,{b,{c,d}}}
{a,{{b,c},d}}
{{a,b},{c,d}}

{{a,{b,c}},d}
{{{a,b},c},d}

One can note that the length of this list is the Catalan number

$$ C_n = \frac{1}{1+n}{2n\choose n} $$

Length[f @@ ConstantArray[a, 6]]
CatalanNumber[6 - 1]
WolframAlpha["answer to life the universe and everything"]

42
42
42

Comments

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