For some integer $z \ge 2$ and large integer $n$ and $ t=\lceil \log n\rceil $, what is an approximate value for the following partial binomial sum? $$ \sum_{i=0}^{n-t} \binom{n}{i}z^i .$$

Another related problem for the same parameters is approximating $$ \sum_{i=n/t}^n \binom{n}{i}z^i .$$

Thanks.