Without a magnetic field any changes we do are seemingly temporary. As we make atmosphere it will be torn away by double solar winds.
Double solar winds are the worst, occurring about 15% of the time this occurs when a faster solar wave catches a slower and rolls into one bigger wave.
And these happen frequently, very frequently!
First Edberg and his colleagues identified 41 doubled solar particle waves and solar powerful particle waves from what are called coronal mass ejections from 2007 to 2008 detected by the Advanced Composition Explorer (ACE) spacecraft that monitors the space weather near Earth. Then they identified 36 of the same events hitting Mars in Mars Express data.
The thin atmosphere left on Mars is due to the small amounts protected at each pole by what remains of the magnetic fields, (which still exist at the poles).
Now I hear you asking, why does Venus have a thick atmosphere but Mars does not?
Mars is much smaller than earth and Venus and it is theorized (not definite) that its liquid outer core cooled and solidified, this had two effects on Mars.
Firstly it meant the large volcanoes became extinct, with no liquid core there was nothing left for them to eject. The volcanoes were responsible for the production of a large amount of the gasses released into the atmosphere, much like they were on Earth millions of years ago. Without these, there was nothing replenishing the atmosphere.
Secondly it meant that the magnetic field slowly died off at all places but the poles, which still retain a small magnetic field. This meant that these double solar waves we mentioned before are now ripping whats left of the atmosphere off of mars.
For the sake of Venus, Venus no longer has a magnetic field, despite popular belief, though the solar winds have torn away all of the lighter hydrogen and oxygen layers. It is unknown to me at this time why the solar winds are unable to tear away the remaining gasses though I think this might be due to them being larger and heavier, such as the carbon dioxide and sulfuric acid layers that make up Venus atmosphere today.
So you could theoretically terra-form Mars by producing atmosphere quicker than its torn away, like the volcanoes tried to do, but that will be expensive and highly resource intensive and not really worth it compared to just sticking a dome there and filling that with air.
There was once talk of liquidating the core with a thermo-nuclear explosion but again, were talking very expensive and difficult.
If one could get the core moving again, presuming it has stopped and solidified, then the magnetic field should replenish itself, and an atmosphere would stay put when generated, but the risks and costs are insanely high compared to finding a planet that already has a magnetic field.
The lack of magnetic field wouldn't stop our attempts, but it would reverse them all over time due to these double solar waves.
The lack of field also wouldn't stop radiation, so any life, such as trees, would die very quickly when exposed to that un-absorbed level.
So yes, terra-forming Mars is theoretically possible, but practically improbable. You are more likely to find bio domes on atmosphere-less planets until we find a simple way to regenerate the magnetic fields.