= Solution
Put
$$
q=p_0+p_1=1-p.
$$
The length process $L_n=\ell(X_n)$ is a nearest-neighbour chain on the nonnegative integers: away from zero it moves up with probability $q$ and down with probability $p$, while at zero it moves up with probability $q$ and stays put with probability $p$.
Returns of the original chain to the root are exactly returns of $L_n$ to zero, so their recurrence classifications agree. The <reflected biased random walk on the nonnegative integers> is transient when $q>p$, null recurrent when $q=p$, and positive recurrent when $q<p$. Since $p=1-q$, the conditions are respectively
$$
\begin{array}{c|c}
\text{classification}&\text{condition}\\ \hline
\text{transient}&p_0+p_1>\tfrac12,\\
\text{null recurrent}&p_0+p_1=\tfrac12,\\
\text{positive recurrent}&p_0+p_1<\tfrac12.
\end{array}
$$
Irreducibility transfers the classification from the root to every state.
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