OpenStreetMap provides information about the type of forest. The information is stored in the tag leaf_type, which has the following values:
Besides this tag, you might also want to check out leaf cycle.
You can use the OverpassAPI to search for a certain type of forest. Using the web interface Overpass ...
If the level of detail in printed topo maps satisfies you, then Outdooractive Pro seems to cover that, at least for some countries. It claims to have Swisstopo, Kompass, DAV/ÖAV, as well as a range of "digitalized official topo maps" which include DTK, ÖK, IGN, OS, Harvey, and some for Slovenia and Italy which I cannot really judge.
I haven't ...