Let $T$ be a rooted Galton-Watson random tree generated accordingly to a probability distribution $\mu$. Now assign to each edge $e$ a random non-negative weight $w_e$ distributed a accordingly to a distribution $\nu$. We also assume that the weights are independent for different edges.

Let $T_{n}$ be the collection of nodes at (hop) distance $n$ from the root. For each $v\in T_{n}$, let $P_{v}$ denote the path from the root to $v$. Define

$$ Z_{v}=\sum_{e\\,\in P_{v}}{\\,w_e}. $$

Now for each $n$ let $Y_{n}=\min_{v\in T_{n}}Z_{v}$. It was proved in Limit distributions for minimal displacement of branching random walks that the sequence of random variables $$ \{Y_{n}-\mathbb{E}(Y_{n})\}_{n\geq 1} $$ is tight.

My question are:

Is it known what is the behavior of $\mathbb{E}(Y_{n})$ as $n$ increases in terms of $\mu$ and $\nu$?

Is it known for the case $\mu=\delta_{k}$, i.e. when $T$ is a $k+1$ regular tree?