How does pg index page optimize dead tuples?

As far as I know, when a index page is full, if you insert a new tuple here, you will split it into two pages.

But pg won't delete the half tuples in the old page in real. So if there is another tuple inserted into this old

page, will pg split it again? I think that's not true, so how it solve this one? please give me a code example,thanks.