Like all other above have said a remote shutter is a must. now what i would do different if i were shooting this picture is i would skip using the flash. instead i would use my cellphone torch or a led light to draw the tree in while capturing on a bulp mode for about 40 seconds. or you could bump up the iso and shoot at shorter time 20- 25 perhaps.
I agree with this, except the bit about the 40 seconds – such a long exposure would likely give you startrails, the length of which being determined by your focal length (the longer the focal length, the less exposure time required for the trails to show).
When shooting the night sky, you really want to go with the largest aperture you can afford. Although not ideal (I'd say f/2.8 or faster would be), f/3.5 could still get the job done depending on the circumstances. f/5 is definitely too slow unless it's a bright night. Set your focus at infinity (or slightly less), and set your ISO at the largest tolerable number (what's tolerable varies by camera and by person). When shooting the night sky I start by setting my camera at f/2.8, ISO 800 and 30 seconds (at 14mm on full frame). If f/3.5 was the fastest I could go, then I'd set my starting ISO at 1250 or 1600, and stick with the 30 seconds. I usually take a couple test shots with those settings and then I make adjustments as needed – if I need more light, I increase the ISO; if I need less light, I stop the aperture down.
I never use flash – if I need to light something in the foreground, I use a flashlight to lightpaint. The problems with using the flash are reach and also that it will likely light something that you may not want to be lit (in this case, the snow close to the camera).
Do you have any photos of this scene shot without the flash (i.e., the tree being just an outline against the sky?).