Yes, the add botton makes the hole jpg image transparent, more or less. If you want more control over the transparent part you should make an targa image of your tree with a mask that defines what should be transparent. Then you use the alpha button.
Ok, I’ve made an alpha mask of my picture.
The tree is now white, and the background is black.
Can I apply the previous texture (the “green” tree with black background) and the new alpha mask on the plane ?
Or do I have to make another picture with an tranparent background ?
Yes, you should make a new picture with a transparent background. UV-map the image and activate the alpha button in the paint buttons menu. Yet it should be transparent. Semitransparency is also possible if you have areas in the alpha layer that only have e.g. 50% opacity. Check the docu of your image manipulation programm to know how to create targas with an alpha layer.
Blender will transform it to a 8bit color depth. The best would be if you reduce the color. Then you have not only more control over the color, it also is faster.