Mmmmm, yes, you can DEFINITELY improve your balance by working on it, and we all definitely lose the ability when we don't work on it.
Most of the balance programs are directed toward seniors, but why wait until poor balance is literally life-threatening, when we can prevent it earlier in life? Anyhow this site has descriptions and illustrations of some basic balance exercises. Tree pose takes some hip flexibility, and Mountain is a little advanced for a beginner IMO.
A couple of things the site doesn't mention, but I find helpful: engage your lower abdominal muscles - imagine zipping up a tight pair of pants. Then imagine that you're hanging from the ceiling by a string attached to the crown of your head. Those two moves will help bring your spine into neutral, where your body is naturally balanced.
In the beginning, you'll want to find a steady focal point for your eyes. Something on a wall in front of you, between waist and eye level. Don't look in a mirror, as this can throw you off. As you progress, you'll advance to doing the exercises with your eyes closed, and looking at moving objects.
Hope the knee's feeling better! HTH!
Speed comes from what you put behind you. - Judi Ketteler