On the independence number of some random trees

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.