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What is the difference between complements and adjuncts? I always have a problem drawing a tree diagram for the syntax structure of a sentence with placing complements with word level category and adjuncts with different nodes; I don't actually know how to distinguish both and where to put them.

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The distinction is between arguments (sometimes also called complements) and adjuncts. In general, arguments are expressions that complete a predicate, and that are required by the predicate. Adjuncts, on the other hand, are not required by the predicate, but they do add (usually temporal or locative) information.
Here are some examples:

(1) Paul lives in London. 
(2) Paul met Peter in London.  

Paul in (1-2) is the subject of the verbs lives in (1) and met in (2). Subjects are arguments. Peter is the object of the verb met in (2). Objects are arguments, too. The PP in London is an argument in (1), because without a locative PP lives would mean something else. In (2), however, in London can be omitted without affecting the meaning of the predicate, hence the PP in (2) is an adjunct.
Look at this page on Wikipedia for more information.

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  • Is there really that much difference? "Paul sat on a chair" = "Paul sat, and the sitting was done on a chair" – amI Oct 9 '17 at 20:04
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Adjuncts are invariably interpreted as nth-order predicates of heads of different ontological types (for n > 1; a first-order predicate is never a syntactic adjunct, it is a syntactic 'predicate'), whereas complements are interpreted as arguments of the predicates that constitute their heads. That holds whether the head is a noun or expanded noun, a verb or expanded verb, an adjective, a few adjective-derived adverbs that do take a complement, or, in fact, any other category, including 'functional' ones.

If, on the contrary, you try to distinguish complements from adjuncts by syntactic criteria, under current Larsonian/Cinquean 'all-in-spec' approaches there is no way to do so, and criteria like optionality/obligatoriness (or inclusion of complements in phrases substituted by pro-forms like one or do so) are notoriously problematic, as well, as certain complements are arguably 'optional', whereas would-be adjuncts like very well in e.g. They treated us very well are obviously obligatory.

Depending on your syntactic assumptions, of course, there may be 'locality' criteria that also distinguish complements from adjuncts, but in surface structure such locality constraints may be cancelled by 'displacement' due to information-structure or by related PF constraints. In general, the above-mentioned semantic criterion is, as far as I know, the easiest and most consistent heuristic tool in this respect.

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  • Re: "would-be adjuncts like very well in e.g. They treated us very well are obviously obligatory": Isn't very well a complement here? – ruakh Mar 31 '16 at 15:33
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Adjuncts and complements are different. An adjunct is not necessary, and adds extra information. A complement is necessary in order to complete the meaning:
[S]He [V]put [O]some salt [C]in the soup. The verb put must have a complement saying where something is put. Without the complement (in the soup), the clause would not be complete. We cannot just say He put some salt.

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This webpage offers another explanation. http://www.ucl.ac.uk/internet-grammar/phfunc/compare.htm

  1. Complements mostly immediately follow the head. They bear a much closer relation to the head than adjuncts
    • fond [(Complement) of biscuits] [(Adjunct) with coffee]: correct
    • ~*fond [(Adjunct) with coffee] [(Complement) of biscuits]: not sound
  2. Adjuncts are stackable. Complements not so. Although one exception is ditransitive verbs which take two complements.

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