I was watching This video and the structure of the rules is very similar to a tree problem, so I came up with a problem I have no clue how to solve.

Fake Avery is on a series of mountains connected by bridges. He can traverse the mountains as follows:

• Each turn, Avery can traverse to an adjacent mountain.
• If a mountain has one neighbor, the mountain itself is deleted, and he is moved to the only neighbor
• If a mountain has multiple neighbors, a random mountain is deleted (maybe this could be the one he just traversed from, to the left of, etc, possibly even query?)
• If a mountain no longer has neighbors, a new mountain is generated.

Avery can also bridge, but I don't know how that would make sense within the context of this problem.

Possible questions derived from these rules:

What is the least amount of mountains Avery can have after k turns?

What is the greatest number of mountains Avery can have after k turns?

Is it possible to visit at least k mountains?

Say another person is traversing the mountains following these same rules. Is it possible to capture them (namely, is it possible for you two to be on the same mountain, assuming optimal play)?

Is it possible to get from node n to some node m?, how many turns would it take?