Figure 1

Case 1 $(\lambda ,\mu )$. The mean-field approximation to the long time distribution $P\left(k\right)={\lim }_{t\to \infty }\mathrm{Pr}\{K\left(t\right)=k\}$ for the coordination number of a random node, plotted on a logarithmic scale. The upper solid line corresponds to the mean-field approximation for $0⩽\mu ∕\lambda ⩽1$. The points shown as discs give the coordination number distribution for a single realization of a large tree grown above the threshold ($\mu ∕\lambda =0.7$ at $\lambda t=40$, giving 251 964 nodes, of which 74 909 were still living). The dotted line and the lower solid line show the mean-field approximation to the distribution for $\mu ∕\lambda =1.5$ and $\mu ∕\lambda =2$, respectively. The points shown as asterisks correspond to the averaged coordination number distribution (20) for $r=1000$ realizations at time $\lambda t=40$ with $\mu ∕\lambda =2$.Reuse & Permissions

Figure 2

Case 2 $(\lambda k,\mu )$. The mean-field approximation to the long time distribution for the coordination number of a random node, $P\left(k\right)$, plotted on a logarithmic scale. The upper solid curve corresponds to the mean-field approximation for $0⩽\mu ∕\lambda ⩽2$. The points shown as disks give the coordination number distribution for a single realization of a large tree grown above the threshold ($\mu ∕\lambda =0.7$ at $\lambda t=10$, giving 137 200 nodes, of which 89 541 were still living). The broken curves and the lower solid curve show the mean-field approximation to the distribution for $\mu ∕\lambda =2.5$, $\mu ∕\lambda =3$, and $\mu ∕\lambda =4$, respectively. The points shown as asterisks correspond to the averaged coordination number distribution (20) for $r=1000$ realizations at time $\lambda t=40$ with $\mu ∕\lambda =4$.Reuse & Permissions

Figure 3

Case 3: $(\lambda k,\mu k)$. Values of the numerically determined parameter $s_p$ plotted as the ratio $s_p∕\lambda$ vs $\mu ∕\lambda$.Reuse & Permissions

Figure 4

Case 3 $(\lambda k,\mu k)$. The mean-field approximation to the long time distribution $P\left(k\right)$ for the coordination number of a random node; $P\left(k\right)$ is plotted on a logarithmic scale. The upper solid curve is the mean-field approximation for $\mu =0$. The mean-field approximations for $\mu ∕\lambda =0.3$, 0.5, 0.7, 1.0 are the upper four broken curves, with corresponding simulation results for large trees shown as squares, disks, diamonds, and triangles respectively. The simulated trees had 193 508, 153 768, 143 226, 21 039 nodes, with 135 258, 76 874, 43 301, 178 living nodes, respectively, at times from $\lambda t=5.0\text{to}\lambda t=40.0$. The lower two broken curves and the lower solid curve show the mean-field approximation for cases where the tree is certain to die out, where $\mu ∕\lambda =1.2$, 1.5, and $2$, respectively. The points shown as asterisks correspond to the averaged coordination number distribution (20) for $r=1000$ realizations at time $\lambda t=10$ with $\mu ∕\lambda =2$.Reuse & Permissions

Figure 5

Case 4 $(\lambda ,\mu k)$. Values of $s_p∕\lambda$ vs $\mu ∕\lambda$. Note that $s_p=0$ when $\lambda =c\mu$, where $c\simeq 1.556$, corresponding to $\mu ∕\lambda \simeq 0.6427$.Reuse & Permissions

Figure 6

Case 4 $(\lambda ,\mu k)$. The mean-field approximation to the long time distribution $P\left(k\right)$ for the coordination number of a random node; $P\left(k\right)$ is plotted on a logarithmic scale. The upper solid curve is the mean-field approximation for $\mu =0$. The mean-field approximations above the threshold with $\mu ∕\lambda =0.1$, 0.3, 0.5 are the upper three broken curves, with corresponding simulation results for large trees shown as squares, discs and diamonds respectively. The simulated trees had 185 707, 139 273, 102 526 nodes, with 151 043, 67 799, 19 853 living nodes, respectively, at times from $\lambda t=5.0\text{to}\lambda t=40.0$. The lower two broken curves and the lower solid curve show the mean-field approximation to the distribution below the threshold, where $\mu ∕\lambda =0.7$, 1, and 1.5, respectively. The points shown as asterisks correspond to the averaged coordination number distribution (20) for $r=1000$ realizations at time $\lambda t=10.0$ with $\mu ∕\lambda =1.5$.Reuse & Permissions

Figure 7

Scaled ring number distributions ${\sigma }_R{\varphi }_R\left(r\right)$ for all four cases for a variety of death rate to birth rate ratios $\mu ∕\lambda$. The gray broken lines indicate average distributions over 100 trees of 10 000 nodes. Data corresponds to $\mu ∕\lambda =0.1$, 0.3, 0.5, 0.8 for case 1 $(\lambda ,\mu )$ and case 3 ($\lambda k,\mu k)$; to $\mu ∕\lambda =0.1$, 0.3, 0.5, 0.8, 1.0, 1.5 for case 2 $(\lambda k,\mu )$; and to $\mu ∕\lambda =0.1$, 0.3, 0.5 for case 4 $(\lambda ,\mu k)$. The black disks indicate the scaled exact ring number distribution found by Chan et al. [9] in case 1 with no death. The black crosses indicate the mean-field ring number distribution found by the same authors for case 2 with no death.Reuse & Permissions

Figure 8

Scaled path length distributions ${\sigma }_L{\varphi }_L\left(l\right)$ for all four cases for a variety of death rate to birth rate ratios $\mu ∕\lambda$. The gray broken lines indicate average distributions over 100 trees of 10 000 nodes. Values of $\mu ∕\lambda$ considered are as in Fig. 7. The black disks indicate the scaled path length distributions in case 1 with no death obtained by Chan et al. [9] using mean-field techniques. The black crosses indicate the approximate distribution in case 2 with no death from Chan et al. [9].Reuse & Permissions