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$n\rightarrow\infty$,thedistributionof$R_{n}$atlarge$n$isalsoexamined\mathrm{.}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 $|\langle R_{n}\rangle-(1+p)/(1-p)|\sim n^{-1/2}$.