Estimation of the number of one-point expansions of a topology which is given on a finite set

Let $X$ be a finite set and $\tau$ be a topology on $X$ which has precisely $m$ open sets. If $t (\tau)$ is the number of possible one-point expansions of the topology $\tau$ on $Y=X\bigcup\{y\}$, then $\frac{m\cdot(m+3)}2-1\ge t(\tau)\ge2\cdot m+\log_2m-1$ and $\frac{m\cdot(m+3)}2-1=t(\tau)$ if and only if $\tau$ is a chain (i.e. it is a linearly ordered set) and $t(\tau)=2\cdot m+\log_2m-1$ if and only if $\tau$ is an atomistic lattice.