This root first appeared as ᴱ√TYALA “play” in the Qenya Lexicon of the 1910s with the derivative ᴱQ. tyalie “play, game” (QL/49). Forms in the contemporaneous Gnomish Lexicon like G. dairtha- “play” < ᴱ✶daı̯al- hint that the actual primitive form might have been ✱ᴱ√DYALA or ✱ᴱ√DAYALA (GL/29). In a table of vowel mutations added as a revision to the Gnomish Lexicon Tolkien instead had forms like ᴱ✶táliı̯èndǝ > G. teilian indicating a primitive initial t- (PE13/116); similar forms appeared in the Early Noldorin Dictionary of the 1920s (PE13/165).
In The Etymologies of the 1930s Tolkien gave ᴹ√TYAL “play” with derivatives like ᴹQ. tyal-/N. telia- “to play” and ᴹQ. tyalie/N. telien “sport, play, game” (Ety/TYAL). Tolkien continued to use Q. tyalië “mirth” in later writings (PE21/80; PE17/64), but at some point in the 1950s decided that initial palatalized dentals were no longer a feature of Primitive Elvish (PE18/93-94), making the ongoing validity of √TYAL as a CE root doubtful: see the entry on how [[at|[j] was lost after initial dentals]] in the Welsh-like branch of the Elvish languages for further details. For example, in the second version of Tengwesta Qenderinwa (TQ2) from around 1950, Tolkien indicated √TYAL was a Quenya-only root, an elaboration of √TAL:
> In Quenya initial groups ty, ny, ps, ks appeared as developments of ky, ñy, sp, sk. But being established they were used in new formations. Thus Q tyal- “play” as a variation on tal- (not kal!) (PE18/84).
Neo-Eldarin: For purposes of Neo-Eldarin, I think it is best to ignore Tolkien’s decision to remove initial palatalized dentals from Primitive Elvish in order to retain roots like √TYAL for both Quenya and (Neo) Sindarin.
A root in The Etymologies of the 1930s serving principally as the basis for N. brethil “beech” (Ety/BERÉTH). In the notes from the late 1950s or early 1960s Tolkien instead connected brethil to the root √BARATH and S. bereth “queen”, where it had the true meaning of “princess” (PE17/23). But that could be a later association rather that a genuine etymology, and doesn’t necessarily invalidate ᴹ√BERETH (especially given the other phonological issues with S. bereth < √BARATH).