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.
"On the independence number of some random trees." Electron. Commun. Probab. 25 1 - 14, 2020. https://doi.org/10.1214/20-ECP345