  We show that for many models of random trees, the independence number divided
by the size converges almost surely to a constant as the size grows to
infinity; the trees that we consider include random recursive trees, binary and
$m$-ary search trees, preferential attachment trees, and others. The limiting
constant is computed, analytically or numerically, for several examples. The
method is based on Crump-Mode-Jagers branching processes.
