Table 1. Key formulas needed for the Applications section.

These approximations will be valid for small transition rates. The approximation for t_1/2 assumes a large initial number of cells at the root, z ≫ 1. For small z see Corollary S2 in S1 Appendix.

Description	Formula	Reference
Weight of path p	$w\left(p\right)=\nu (p_1,p_2){\prod }_{i=2}^l\frac{\nu (p_i,p_{i+1})}{\lambda -\lambda \left(p_i\right)}$	Eq (2)
Total weight of target	${\varphi }_N={\sum }_{p\in {\mathcal{P}}_{1,N}}w\left(p\right)$	Eq (3)
Distribution of target hitting time	$\mathbb{P}(T>t)\approx {\left(\frac{\lambda /\alpha }{1+e^{\lambda t}{\varphi }_N\alpha /{\lambda }^2}+\beta /\alpha \right)}^z$	Eq (5)
Median of target hitting time	$t_{1/2}\approx \frac{1}{\lambda }\text{log}\frac{\lambda }{z{\varphi }_N}$	Eq (7)
Probability target populated via path p	${\mathbb{P}}_1(T(p)=T)\approx \frac{w(p)}{{\varphi }_N}$	Eq (10)