= Solution
The open descendants of any vertex form a <Galton-Watson process> with offspring distribution $\operatorname{Bin}(k,p)$. If $kp\leq1$, it dies out almost surely, so $N_\infty=0$.
If $kp>1$, let $\theta>0$ be its survival probability. At level $n$, each vertex begins an independent descendant subtree; the event that its edge to its parent is closed while its descendant open cluster is infinite has probability $(1-p)\theta>0$. Thus the number of such vertices at level $n$ is binomial with $k^n$ trials and a fixed positive success probability. For every fixed $M$, the probability of at least $M$ successes tends to one. Their infinite clusters are separated by their closed parent edges, so $\mathbb P(N_\infty\geq M)=1$ for every $M$, and hence $N_\infty=\infty$ almost surely.
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