The Resistance Of Randomly Grown Trees

An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability $p$ or two edges with probability $1-p$. With each edge having a resistance equal to 1, the total resistance $R_{n}$ between the root vertex and a busbar connecting all the vertices at the $n^{th}$ level is considered. Representing $R_{n}$ as a dynamical system it is shown that $\langle R_{n} \rangle$ approaches $(1+p)/(1-p)$ as $n\rightarrow\infty$, the distribution of $R_{n}$ at large $n$ is also examined. Additionally, expressing $R_{n}$ as a random sequence, its mean is shown to be related to the Legendre polynomials and that it converges to the mean with $|\langle R_{n}\rangle-(1+p)/(1-p)|\sim n^{-1/2}$.