It is hard to say exactly how long we would last without trees. In the short term, we could certainly survive. Trees are often considered valuable, because they produce oxygen during photosynthesis, but most of the oxygen in the atmosphere is produced by phytoplankton in the ocean.

Without trees, we would still have enough oxygen to survive. The bigger problem of not having trees is that we need trees to capture and store carbon. Without trees, climate change would happen much faster and put more people in danger. Trees also remove other pollutants from the air, are an important part of the water cycle, and provide food, building materials, and animal habitat. So we do need trees, but how long we would last without them depends on many other factors.

Maybe 4000 years...
"A WORLD WITHOUT TREES
Would it be sufficient for humans to survive? In one year, a mature leafy tree produces as much oxygen as ten people breathe. If phytoplankton provides us with half our required oxygen, at current population levels we could survive on Earth for at least 4000 years before the oxygen store ran empty.Apr 22, 2016 Source.

This is difficult to answer. Humans need oxygen to breathe, and trees produce oxygen, but trees are not the only plants that produce oxygen, so theoretically, humans don't need trees to survive at all. However, trees help the planet in numerous ways, and they do produce a great deal of oxygen, so not having trees would surely be bad for us in the long run - it's just hard to say how long because there are so many other questions.