Scuba_Dave

What is going to be growing on the trellis ?
How long do you expect/want to use it ?
More important then the weight is the effect of wind on the structure once the vine etc grows out
Once the structure has vines on it then the wind force increases & in many cases will blow it over
If you only want to use it for 1 growing season then what you built may work
But a 40-50 mph wind may blow it over

If the 1x2's are reg wood then they will not last long, the grade of wood used to make them is not that great
Also a single nail in each end of the 1x2 shown in the pic will not prevent the structure from moving/swaying
The "X" bracing will greatly increase the strength of the structure
"Fresh" built most things seem stable, its once wind/water & sun start taking their toll that the wood/structure will start to move as time goes on

I use PT 4x4's for (almost) all uprights when I build something, or larger - for my deck
One exception being my greenhouse -but its attached to the house & bolted to a cement foundation