Beech Reading

NIL 2-4 treats the PIE Etymon *b^hah₂g-ó- "beech". They mention that some scholars reconstruct long /a:/ and some (not always the same) scholars link it to the previously discussed *b^hag-. In general I don't see any reasonable link between a tree name and a root meaning "share" etc. But there is a possible connection for the Germanic cognates meaning "book, letter".
NIL also mentions (FN 2 on p.3) that there are doubts about the relationship between the words meaning "book, letter" and the continuants of this root meaning "beech". The main formal problem is that the words meaning "letter / rune" seem to go back to a root noun, that actually is attested in Old English, while the beech words are eh₂ stems (or n-stems derived from them). So we have derived morphology for what is supposed to be the original meaning ("beech") and a root noun for what is supposed to be the derived meaning ("book, letter").
Elmar Seebold, addressing this issue in his "Etymologie: eine Einführung am Beispiel der deutschen Sprache", (München : Beck 1981, quoted herafter as "Et."), also mentions that the proposed writing on beech tables that is supposed to be behind the change of the meaning from "beech" to "letter" is actually not attested, neither archaeologically, nor in written sources; Germanic runes are attested only on bark, stone, and various household objects (Et., pp. 290-291). It is also clear that the original meaning was "letter", not "book" - the oldest attestations mean "letter" in the singular and "document, book" in the plural, an obvious calque from Latin (littera - litterae) and Greek (gramma - grammata) (Et. p. 290). Seebold argues that the meaning "letter" is derived from the compound Norse bókstafr, Old Saxon bōkstaf, OHG. buohstap, whose second member means "staff". The writing of runes on staffs is widely attested (here Seebold is undermining his previous argument somewhat, as these staffs could of course have been made of beech wood, but the written source he quotes actually mentions ash wood).  
He then adduces a parallel from Welsh, coelbren "sign-wood, lot-wood", composed of coel "sign, omen" and bren "wood", designing a piece of wood covered with signs used to throw lots, a custom also attested for the Germanic people. He takes this parallel as an indication that the first element of bókstafr etc. originally meant "sign, omen, lot", and links it with our old acquaintance, the root *b^hag-, reconstructing a root noun Proto-Germanic *bōk-s "lot, portion" (Et. pp. 291-292). That would mean that the word family of "book" is not related to "beech", but that the purported writing on beech tablets is only a folk etymology. (A shorter version of these arguments can also be found in Kluge(-Seebold), "Etymologisches Wörterbuch der deutschen Sprache", s.v. Buch).
In general, I find this argumentation attractive. A formal problem is that the proposed root noun is attested only in Vedic, as a second part of compounds, with an active meaning "enjoying" (NIL p.1), but not as an independent word with a resultative meaning "allotment, lot", and that it would be the only continuation of *b^hag- in Germanic. The same problems woud also arise if, as I proposed,  we eliminate the root b^hag- and take its purported continuations as derived from the root b^heg- "break"; that root also is not continued in Germanic (at least according to LIV p. 66/67 and NIL, p. 6; the forms with nasal infix mentioned in IEW p.115 look onomatopoetic, for which reason Pokorny himself states that they don't belong to *b^heg- ). There is also no root noun formed from *b^heg- attested in NIL (p. 6); if we eliminate *b^hag-, we would have at least the Vedic root noun mentioned above, but still with the same problems. On the other hand, an isolated root noun from a root that otherwise doesn't have any cognates in Germanic is a perfect candidate for the kind of folk etymology discussed.   

PIE *bhag- and Armenian bak

This is a follow-up on my thoughts on PIE *b^hag. I’ve come across an article by Hrach Martirosyan (“The place of Armenian in the Indo-European Language family: the relationship with Greek and Indo-Iranian”, Journal of Language Relationship, No. 10 / August 2013, p. 85 - 138), PDF here, where he adduces Armenian bak “courtyard; sheep pen, sun or moon halo” (missing in NIL) as a cognate of Indo-Iranian *b^ha:gá-: Sanskrit bha:gá- m. “prosperity, good fortune, property, personified distribution”, Old Avestan ba:ga- “part”, the descendants of which took on the meaning “landed property, fief, garden” (p.99, §5.1.3). Martirosyan admits the possibility that this is not a cognate, but an old loan from Iranian; he names one argument for it being a loan, namely the fact that the Armenian word is an a-stem, while the Indo-Iranian correspondences are o-stems; incorporation as an a-stem seems to be the expected outcome for an Iranian *ba:ga-; as another argument for a loan I would also see the fact that there seem to be no other formations from a root *b^hag- in Armenian. On the other hand, it would have to be an old loan from before the Armenian consonant shift, but Martirosyan admits that there are other such loans.

If this is not a loan, but a cognate, it would require a proto-form *ba:g-a:-, which could be explained as a Vrddhi-formation from *b^hag- or point to a PIE *b^heH₂g-eH₂- (Martirosyan’s reconstruction). Therefore, accepting bak as a cognate would in any case require us to posit a root PIE *b^hag- or b^heH₂g- separate from *b^heg- “break” (continuants of the latter root are well-attested in Armenian).