Europe’s mountain forests, which are naturally valuable areas due to their high biodiversity and well-preserved natural characteristics, are experiencing major alterations, so an important component of monitoring is obtaining up-to-date information concerning species composition, extent, and location. An important aspect of mapping tree stands is the selection of remote sensing data that vary in temporal, spectral, and spatial resolution, as well as in open and commercial access. For the Tatra Mountains area, which is a unique alpine ecosystem in central Europe, we classified 13 woody species by iterative machine learning methods using random forest (RF) and support vector machine (SVM) algorithms of more than 1000 polygons collected in the field. For this task, we used free Sentinel-2 multitemporal satellite data (10 m pixel size, 12 spectral bands, and 21 acquisition dates), commercial PlanetScope data (3 m pixel size, 8 spectral bands, and 3 acquisitions dates), and airborne HySpex hyperspectral data (2 m pixel size, 430 spectral bands, and a single acquisition) with fusion of the data of topographic derivatives based on Shuttle Radar Topography Mission (SRTM) and airborne laser scanning (ALS) data. The iterative classification method achieved the highest F1-score with HySpex (0.95 RF; 0.92 SVM) imagery, but the multitemporal Sentinel-2 data cube, which consisted of 21 scenes, offered comparable results (0.93 RF; 0.89 SVM). The three images of the high-resolution PlanetScope produced slightly less accurate results (0.89 RF; 0.87 SVM).