When we talk about tree search we allow for backtracking, so if a node has 3 children all 3 will be explored generally, or at least a subsample of the children will be, in LLM sampling you generally pick a single token/child and then just go on with that until the end of the generation.
If DeepMind is indeed doing something similar to AlphaZero to language modelling one would expect they would generate multiple "rollouts" from the current context and then use some kind of function/network to predict which next token will lead you to the best final generation and then output that token. How to do all of that using a sensible amount of compute is what remains to be seen