I have a balanced tree $T(V,E)$ of constant degree $d$. I remove each vertex from the tree with equal probability $p$. I refer to the largest remaining tree with the same root as $T'(V',E')$ (this tree can potentially be empty in the case that the root of $T$ was removed). I am trying to solve the CDF $P(|V'| > n)$.

I was able to solve the expectation $E[|V'|]$ using the observation that the expectation of the number of vertices in a tree with root $r$ is $E[|T_r|] = (1-p)(1 + \sum_{v \in children(r)}E[|T_v|])$. I have not found a way to solve the CDF.