Is anyone aware of any attempts to count or approximate the number of trees there are in Sweden or Scandinavia? I have a small popular science side project for which this would be useful
I would definatley look at data from Swedish National Forest Inventory (Riksskogstaxeringen). They have time series data of multiple variables (DBH, size classes, species, dead wood etc) from plots all over the country. Extrapolations based on these data should produce reasonable estimates of total tree count, and probably the most accurate one that can be obtained at the moment. As an example, this report (in swedish) includes graphs on trees/km$^2$ for different species and areas (page 17 ff), but this is only based on areas that can support productive forestry.
Did a bit of a search, but could not find any Scandinavian specific statistics; however, I did find European Commission Forestry Statistics page, within this site there are some stats by country. Further, the website About Forestry in Northern European countries also has a few links to country based data.
I hope this helps.