Let $T$ be a complete binary tree with $2^n$ leaves and let $I$ be a random variable on $\{0,1,...,n-1\}$ that outputs $i$ with probability $p_i$ for each $i=0,...,n-1$.
Two nodes are siblings if and only if they share a parent. The sibling of a node's parent is called that node's 1-uncle; the sibling of its grandparent is that node's 2-uncle; the sibling of its great-grandparent is its 3-uncle, and so on. For a given leaf node $l$, the uncle set of $l$ consists of all its $i$-uncles for $i=1,\ldots,n-1$.
Now, consider the following experiment: In each round, toss an unbiased $2^n$-sided die to get a leaf index $l$ and sample an $i\in\{0,\ldots,n-1\}$ using $I$. Mark both the $l$th leaf and (if $i>0$) that leaf's $i$-uncle as "known". Finally, mark any node for which both children are "known" as being "known" (recursively until no new nodes become "known").
Fix $k\in\{1,\ldots,2^n\}$. What is the expected number of rounds before the $k$th leaf and every element of its uncle set are "known"?
(I am also interested in the answer in the case where we mark every uncle from the $1$-uncle up to the $i$-uncle as "known" in each round.)