To build a log deer that will last for a while, you should consider building it so that you can disassemble it, store it, and get it back out for future Holidays.
I have seen people screw or nail the legs on, and the neck and head. People with an abundance of logs can do that. They can just make another one.
If you want to build one to last, I found that log tenon and mortice joinery works best. It also allows you to twist a leg (not break a leg), turn a head, a neck, and adjust the deer to different positions. If you have a couple of deer, you can twist their head to be looking at each other, or something else, and change it any time you want. It is great for kids. It is like a big puzzle, putting them together. And, just think of the fun in decorating them with red noses, ribbons and bows, maybe even a little sweater or something.
You can use trees with brown bark, or a tree that has white bark like birch or poplar. If you keep your deer outside, they will weather and the bark will eventually fall off. If you keep them indoors, they will last forever. With a tenon and mortise joint, you simply take them apart for storage until next year.
With the E-Z Log Tenon Cutter, you can do that. You are only limited to about a 4" diameter log to create a 1" up to a 3" diameter tenon. If your log is a little larger than 4" in diameter, just trim it down with a chain saw, planer, or hand saw so it fits the E-Z Log Tenon Cutter.
It is probably too late for this year, but one of these days I will post plans to build log reindeer. You can use your imagination to create what you want. Some deer I've seen are really out of proportion, but that is the beauty of it. Long legs. Short Legs. Big ears. Skinny bodies. It's all fun!!!!