You're always pushing new elements on the same array in different positions, not different depths:

push @{$list[$depth]}, @current_list;
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For instance, in order to h2 being a child of greybox you should:

push @{$list[0][3]}, @current_list;
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Where depth means there's 2 array references involved. That's why the recursive solution provied by jethro works better: recursion and depth are intimate concepts.