Indeed, if the initial T in TMR is radical, then it appears to me to be of the family of roots CMR (of which we have CAMERET צמרת 'tree top'), SMR (of which we have MA-SMER 'nail'), SHMR (of which we have SHAMIR שמיר).
Otherwise, the initial T-, or TA-, is possibly the attached personal pronoun ATAH (אתה) which is used to turn the root MR into the name TA-MAR ('you are MAR'). I don't think that this MAR is here in the sense of 'bitter', but is rather here merely a short form of AMIR (אמיר 'upper branches').
Notice that several tree names in the TANAK have this dual form:
TAMAR - MOR
TIRZAH (TIRCAH?) -EREZ
TOREN - OREN
TA'ASHUR - ASHERAH
We also find the tree names:
T'ENAH
TAPUAK
TIDHAR
Isaac Fried, Boston University
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