We present N/O abundance ratios, ionization parameters q ion , and oxygen abundance O/H for a total of 41 galaxies (11 individual galaxies and a 30-galaxy stack) including Lyα emitters and Lyman break galaxies at z ∼ 2, and investigate galaxy evolution from z ∼ 0 to 2 in conjunction with 208,529 local SDSS galaxies and 9 green pea galaxies (GPs). In contrast with most of the recent studies, we obtain the N/O ratio, q ion , and O/H measurements by direct T e methods with [O III] λ4363 and O III] λ1665 lines. Based on these reliable measurements, we find that there exist z ∼ 2 galaxies with an excess of N/O falling beyond the local average of N/O-O/H relation, while the majority of the z ∼ 2 galaxies have the N/O ratios nearly comparable with z ∼ 0 galaxies in the N/O-stellar mass relation. Our galaxies place the upper limit of N/O ratio log(N/O) ≤ −1.26 on average, suggesting that the N/O ratio evolves, if at all, by < 0.17 dex. Exploiting our reliable measurements free from the N/O-q ion -O/H degeneracies, we identify, for the first time, that z ∼ 2 galaxies with offsets in the BPT diagram indicate either 1) only an N/O excess, 2) only a q ion excess, or 3) both N/O and q ion excesses. We argue that the BPT offsets at z ∼ 2 are not made by one of 1)-3) galaxy populations alone, but the composite of 1)-3) populations. We confirm that these 1)-3) populations also exist at z ∼ 0, such as GPs and SDSS low-mass and high-SFR galaxies (LMHSs).