One A.C. Nunn asked Tolkien much the same question in 1958-9. Part of his letter reads:
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Therefore, one of the following must be true: (1)Sméagol's people were not "of hobbit-kind" as suggested by Gandalf (I. p. 62); (2) the Hobbit custom of giving presents was a recent growth; (3) the customs of the Stoors differed from other Hobbits; or (5) [sic] there is an error in the text.
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Tolkien responded with a very full exploration of the whole subject.
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In the matter of birthday-customs and apparent discrepancies that you note, we can therefore, I think, dismiss your alternatives (1) and (5). You omit (4).
With regard to (1) Gandalf certainly says at first 'I guess' p. 62; but that is in accordance with his character and wisdom. In more modern language he would have said 'I deduce', referring to matters that had not come under his direct observation, but on which he had formed a conclusion based on study... But he did not in fact doubt his conclusion 'It is true all the same, etc.' p. 63.
Your alternative (2) would be possible; but since the recorder says on p. 35 Hobbits..., and not the Hobbits of the Shire, or Shire-folk, it must be supposed that he means that the custom of giving presents was in some form common to all varieties, including Stoors. But since your (3) is naturally true, we might expect even so deep-rooted a custom to be exhibited in rather different ways in different branches. With the reemigration of the Stoors back to Wilderland in TA 1356, all contact between this retrograde group and the ancestors of the Shirefolk was broken. More than 1100 years elapsed before the Déagol-Sméagol incident (c. 2463). At the time of the Party in TA 3001, when the customs of the Shire-folk are cursorily alluded to insofar as they affect the story, the gap of time was nearly 1650 years.
...the facts concerning the Shire could be set out in some detail. The riverside Stoors must, naturally, remain more conjectural...
With regard to presents: on his birthday the 'byrding' both gave and received presents; but the processes were different in origin, function and etiquette. The reception was omitted by the narrator (since it does not concern the Party) but it was in fact the older custom, and therefore the one most formalised.
Letters #214 (c.1958-9)
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This letter goes on at some length and is well worth reading, but it's far too long for me to post it all here and still get some sleep tonight. Suffice it to say that whatever the situation in
LR when published, Tolkien very rapidly invented an elaborate set of Hobbit birthday-customs that explain away the whole thing.
As for it actually being Sméagol's birthday, that is quite well-attested. Firstly, in the above letter Tolkien takes it for granted that it was. Secondly, Gandalf also accepts the fact without question:
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'The murder of Déagol haunted Gollum, and he had made up a defence, repeating it to his "precious" over and over again, as he gnawed bones in the dark, until he almost believed it. It was his birthday. Déagol ought to have given the ring to him.
The Fellowship of the Ring, Book I, p. 55
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Of course, the fact of it having been his birthday could well have been another deluded invention, but later in his letter, Tolkien wrote:
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Déagol, evidently a relative (as no doubt all the members of the small community were), had already given his customary present to Sméagol... Being a mean little soul he grudged it. Sméagol, being meaner and greedier, tried to use the 'birthday' as an excuse for an act of tyranny.
Letters #214
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Therefore, it was Gollum's birthday and Hobbits both gave and received presents on their birthdays. I strongly recommend the letter above to all students of Hobbit customs and etiquette.