Let $T$ be an infinitely rooted binary tree. Let $\Gamma$ be the tree obtained from $T$ by replacing each edge whose verticies are at distances $k$ and $k+1$ from the root by a chain of $2^k$ edges. Determine whether simple random walk on Γ is recurrent or transient.
I'm not too sure how to think of this. For a binary tree you consider a sequence of random variables corresponding to the level which has probability $\frac{2}{3}$ of moving down and $\frac{1}{3}$ of moving up. Do I still consider the levels here and invoke borel cantelli? It is difficult since the length of the chains are not constant