Hello,
You can perform a supervised classification (such as Random Forest classifier) using SNAP toolbox.
I do not think that using only a SAR image will help you to accurately identify forest species. Due to the high sensitivity of microwaves to surface roughness, all the different forest species produce very similar backscatter signature which makes it very hard for the algorithm classify them correctly. To achieve good results in discriminating forest species, additional datasets are needed.
I would suggest you to use an optical image (e.g SENTINEL-2) and see what you can get from that. I ideally, a high resolution optical image is required (WorldView-2) for being able to identify all the trees in your area of interest
In terms of training the model to perform a classification, Please have a look at the following posts: Rndom forest classification steps
Supervised and unsupervised classification, Sentinel 2