Consider a random walk on the infinite binary tree with root $x$ which has the following transition probabilities. $$ p_{x,0}=p_{x,1}=\frac{1}{2},~~~p_{y,y0}=p,~~~p_{y,y1}=q,~~\text{and }p_{y,\tau y}=r, $$ for $y\in\bigcup_{n=1}^{\infty}\left\{0,1\right\}^n=:\Sigma^*$, where $\tau\colon\Sigma^*\to\Sigma\cup\left\{x\right\}$ is defined by $$ \tau(y_1\cdots y_{n-1}y_n):=y_1\cdots y_{n-1}\text{ for }n\geq 2,~~\tau(0)=\tau(1):=x. $$ Assume that $p,q,r\geq 0$ and $p+q+r=1$.
Determine $p,q,r$ for which the random walk is
(a) reducible,
(b) irreducible and transient and
(c) irreducible and recurrent.
EDIT
I add my new results:
(a) reducible if at least one of $p,q,r$ is $0$.
(b) irreducible and transient for
$$ r\in \left(0,\frac{1}{2}\right)\cup\left(\frac{1}{2},1\right)\text{ and } p,q\in (0,1) \text{ such that }p+q=1-r. $$ .
(c) irreducible and recurrent for $$ r=\frac{1}{2}\text{ and }p,q\in (0,\frac{1}{2}]\text{ such that }p+q=\frac{1}{2}. $$