No - MoE is just a way to add more parameters to a model without increasing the cost (number of FLOPs) of running it.
The way MoE does this is by having multiple alternate parallel paths through some parts of the model, together with a routing component that decides which path (one only) to send each token through. These paths are the "experts", but the name doesn't really correspond to any intuitive notion of expert. So, rather than having 1 path with N parameters, you have M paths (experts) each with N parameters, but each token only goes through one of them, so number of FLOPs is unchanged.
With tree search, whether for a game like Chess or potentially LLMs, you are growing a "tree" of all possible alternate branching continuations of the game (sentence), and keeping the number of these branches under control by evaluating each branch (= sequence of moves) to see if it is worth continuing to grow, and if not discarding it ("pruning" it off the tree).
With Chess, pruning is easy since you just need to look at the board position at the tip of the branch and decide if it's a good enough position to continue playing from (extending the branch). With an LLM each branch would represent an alternate continuation of the input prompt, and to decide whether to prune it or not you'd have to pass the input + branch to another LLM and have it decide if it looked promising or not (easier said than done!).
So, MoE is just a way to cap the cost of running a model, while tree search is a way to explore alternate continuations and decide which ones to discard, and which ones to explore (evaluate) further.