One-arm domination time in Cylindrical Hastings-Levitov$(0)$

The cylindrical Hastings-Levitov$(0)$ admits a single infinite connected tree (arm). For a cylinder of width $N$, and particles of size $\lambda$, we consider the last time ${\upsilon }_{N,\lambda }$, that a finite tree receives a particle. We prove that $\frac{c{N}^2}{{\lambda }^3}<\mathbb{E}[{\upsilon }_{N,\lambda }]<\frac{C{N}^2}{{\lambda }^3}$. Furthermore, we establish an exponential tail for ${\upsilon }_{N,\lambda }$.