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There have been a couple of times in my life when people from multi-level marketing organizations attempted to recruit me. I listened to what they had to say, and both times I did not get involved ...

Consider a rooted tree $T$ and $n$ leaf nodes which are all at depth $R$. We would like to select a random subset $S$ of the edges of $T$, such that (i) Every root-leaf path of $T$ contains at least ...

Let $T_n$ be a random tree on $n$ labelled vertices chosen equiprobably among all $n^{n - 2}$ trees, and $I(T)$ be the number of distinct independent sets of a tree $T$. I'm interested in the average ...

This question follows up on Bound on queries to a tree with unusual probabilities, where @fedja was able to disprove my conjecture under only constraints (1-4) below. I restate the relevant facts here ...

Consider a tree $\mathcal{T}(r) = (V,E)$ rooted at $r \in V$. Let $\kappa_r: V \longrightarrow [0,1]$ such that $\sum_{v \in V} \kappa_r(v)^2 = 1$. Furthermore, for any given vertex $v \neq r$, $\...