  We study the size and the lifetime distributions of scale-free random
branching tree in which $k$ branches are generated from a node at each time
step with probability $q_k\sim k^{-\gamma}$. In particular, we focus on
finite-size trees in a supercritical phase, where the mean branching number
$C=\sum_k k q_k$ is larger than 1. The tree-size distribution $p(s)$ exhibits a
crossover behavior when $2 < \gamma < 3$; A characteristic tree size $s_c$
exists such that for $s \ll s_c$, $p(s)\sim s^{-\gamma/(\gamma-1)}$ and for $s
\gg s_c$, $p(s)\sim s^{-3/2}\exp(-s/s_c)$, where $s_c$ scales as $\sim
(C-1)^{-(\gamma-1)/(\gamma-2)}$. For $\gamma > 3$, it follows the conventional
mean-field solution, $p(s)\sim s^{-3/2}\exp(-s/s_c)$ with $s_c\sim (C-1)^{-2}$.
The lifetime distribution is also derived. It behaves as $\ell(t)\sim
t^{-(\gamma-1)/(\gamma-2)}$ for $2 < \gamma < 3$, and $\sim t^{-2}$ for $\gamma
> 3$ when branching step $t \ll t_c \sim (C-1)^{-1}$, and $\ell(t)\sim
\exp(-t/t_c)$ for all $\gamma > 2$ when $t \gg t_c$. The analytic solutions are
corroborated by numerical results.
