 Solution

From any vertex, following the parent edge repeatedly reaches the root with positive probability $p^k$. From the root, any prescribed binary string $(b_1,\dots ,b_k)$ can be reached by following its bits, with positive probability $p_{b_1}\cdots p_{b_k}$. Thus every state communicates with every other state and the chain is irreducible.

At the root there is a self-loop of probability $p>0$, so the root has period one. All states of an irreducible chain have the same period; hence the chain is aperiodic.