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 qkββΌkβΞ³. In particular, we focus on
finite-size trees in a supercritical phase, where the mean branching number
C=βkβkqkβ is larger than 1. The tree-size distribution p(s) exhibits a
crossover behavior when 2<Ξ³<3; A characteristic tree size scβ
exists such that for sβͺscβ, p(s)βΌsβΞ³/(Ξ³β1) and for sβ«scβ, p(s)βΌsβ3/2exp(βs/scβ), where scβ scales as βΌ(Cβ1)β(Ξ³β1)/(Ξ³β2). For Ξ³>3, it follows the conventional
mean-field solution, p(s)βΌsβ3/2exp(βs/scβ) with scββΌ(Cβ1)β2.
The lifetime distribution is also derived. It behaves as β(t)βΌtβ(Ξ³β1)/(Ξ³β2) for 2<Ξ³<3, and βΌtβ2 for Ξ³>3 when branching step tβͺtcββΌ(Cβ1)β1, and β(t)βΌexp(βt/tcβ) for all Ξ³>2 when tβ«tcβ. The analytic solutions are
corroborated by numerical results.Comment: 6 pages, 6 figure