Return to Answer

Try the code

levelTree[expr_] := Replace[expr, {h_[x___] -> {h, x}}, {0, Infinity}];    
levelTree @ {a, {b1, b2}, {c, {d1, d2}}}    

which returns

{List, a, {List, b1, b2}, {List, c, {List, d1, d2}}}

A simple exmaple

levelTree[a b + c d]

which returns

{Plus, {Times, a, b}, {Times, c, d}}

I like the lispy variation

levelTree[expr_] := Replace[expr, (h : Except[List])[x___] -> {h, x}, {0, Infinity}];
levelTree@cons[car[1, 2], cdr[3, 4]]

which returns

{cons, {car, 1, 2}, {cdr, 3, 4}}

Try the code

levelTree[expr_] := Replace[expr, {h_[x___] -> {h, x}}, {0, Infinity}];    
levelTree @ {a, {b1, b2}, {c, {d1, d2}}}    

which returns

{List, a, {List, b1, b2}, {List, c, {List, d1, d2}}}

A simple exmaple

levelTree[a b + c d]

which returns

{Plus, {Times, a, b}, {Times, c, d}}

I like the lispy variation

levelTree[expr_] := Replace[expr, (h : Except[List])[x___] -> {h, x}, {0, Infinity}];
levelTree @ plus[car[{1, 2}], cdr[{3, 4}]]

which returns

{plus, {car, {1, 2}}, {cdr, {3, 4}}}

Given any of these results, you can now use TreeForm[] or ExpressionGraph[] or some other custom Graph display.

Try the code

levelTree[expr_] := Replace[expr, {h_[x___] -> {h, x}}, {0, Infinity}];    
levelTree @ {a, {b1, b2}, {c, {d1, d2}}}    

which returns

{List, a, {List, b1, b2}, {List, c, {List, d1, d2}}}

A simple exmaple

levelTree[a b + c d]

which returns

{Plus, {Times, a, b}, {Times, c, d}}

Try the code

levelTree[expr_] := Replace[expr, {h_[x___] -> {h, x}}, {0, Infinity}];    
levelTree @ {a, {b1, b2}, {c, {d1, d2}}}    

which returns

{List, a, {List, b1, b2}, {List, c, {List, d1, d2}}}

A simple exmaple

levelTree[a b + c d]

which returns

{Plus, {Times, a, b}, {Times, c, d}}

I like the lispy variation

levelTree[expr_] := Replace[expr, (h : Except[List])[x___] -> {h, x}, {0, Infinity}];
levelTree@cons[car[1, 2], cdr[3, 4]]

which returns

{cons, {car, 1, 2}, {cdr, 3, 4}}

Try the code

Replace[expr, {h_[x___] -> {h, x}}, {0, Infinity}]

which returns

{List, a, {List, b1, b2}, {List, c, {List, d1, d2}}}

Try the code

levelTree[expr_] := Replace[expr, {h_[x___] -> {h, x}}, {0, Infinity}];    
levelTree @ {a, {b1, b2}, {c, {d1, d2}}}    

which returns

{List, a, {List, b1, b2}, {List, c, {List, d1, d2}}}

A simple exmaple

levelTree[a b + c d]

which returns

{Plus, {Times, a, b}, {Times, c, d}}