Sorry if this question is too easy to answer

I have a growing tree. It starts with a little square and I want to make little branches to come out of it and create a huge tree with lots of branches all interlaced. I imagine that I was able to start with the square in second 0, go to second 1 and add some points and stretch them out to create a little branch, then add more points and make the "thing" grow. To my surprise, all the points I added at second one were also added at frame 0, thus I have no square at second 0 no more, but lots of misplaced points.

This makes me wonder: what if I want new points to be created in the middle of the animation? That is, I want to have a square at second 1 and a pentagon at second 2 by adding a point. How can you make that?
BTW, when doing the branches, All the branches were one shape (one fill), so when one branch passed over the other, the intersection became blank (transparent). I can make that using two shapes, but if I need to overlap one part of the shape with itself, is there a way to avoid that "erase" effect ?
Thanks!
William.