Binary rooted tree isomorphism
There is a classical linear time algorithm for rooted tree isomorphism due to Aho, Hopcroft and Ullman. The algorithm actually uses a simple isomorphism invariant. See for example lecture notes of Vikram Sharma. Using this, you can solve unrooted tree isomorphism in linear time, as described for example in Smal's slides. Another classic algorithm is due to Buss.