  We study several parameters of a random Bienaym\'e-Galton-Watson tree $T_n$
of size $n$ defined in terms of an offspring distribution $\xi$ with mean $1$
and nonzero finite variance $\sigma^2$. Let $f(s)={\bf E}\{s^\xi\}$ be the
generating function of the random variable $\xi$. We show that the independence
number is in probability asymptotic to $qn$, where $q$ is the unique solution
to $q = f(1-q)$. One of the many algorithms for finding the largest independent
set of nodes uses a notion of repeated peeling away of all leaves and their
parents. The number of rounds of peeling is shown to be in probability
asymptotic to $\log n / \log\bigl(1/f'(1-q)\bigr)$. Finally, we study a related
parameter which we call the leaf-height. Also sometimes called the protection
number, this is the maximal shortest path length between any node and a leaf in
its subtree. If $p_1 = {\bf P}\{\xi=1\}>0$, then we show that the maximum
leaf-height over all nodes in $T_n$ is in probability asymptotic to $\log
n/\log(1/p_1)$. If $p_1 = 0$ and $\kappa$ is the first integer $i>1$ with ${\bf
P}\{\xi=i\}>0$, then the leaf-height is in probability asymptotic to
$\log_\kappa\log n$.
