Let $\T_(t,n)$ be a continuous-time critical branching process conditioned to have population $n$ at time $t$. Consider $T_(t,n)$ as a random rooted tree with edge-lengths. We define the genealogy $\G(T_t)$ of the population at time $t$ to be the smallest subtree of $\T_(t,n)$ containing all the edges at a distance $t$ from the root. We also consider a Bernoulli($p$) sampling process on the leaves of $\T_(t,n)$, and define the $p$-sampled history $\H_p(\T_(t,n))$ to be the smallest subtree of $\T_(t,n)$ containing all the sampled leaves at a distance less than $t$ from the root. We first give a representation of $\G(\T_(t,n))$ and $\H_p(\T_(t,n))$ in terms of point-processes, and then prove their convergence as $n\rightarrow\infty$, $\frac(t)(n)\rightarrow t_0$, and $np\rightarrow p_0$. The resulting asymptotic processes are related to a Brownian excursion conditioned to have local time at $0$ equal to $1$, sampled at times of a Poisson($\frac(p_0)(2)$) process.