On trees: Can you use regular references for the children, and weak references for the link back to the parent? That's a valid structure that tears down properly. With strong backpointers, when one of the objects is deleted, there's a moment when there's still a live reference to it. That's invalid under Rust's rules, so you can't do that. There's also the option to own all tree objects with some collection for alloc…
He only way for Rust's type system to understand these kinds of access patterns in a way that doesn't require the insertion of "shared mutability" types (e.g. Mutexes) involves mutating the tree during traversals.
A simple version of such a pattern: a "doubly linked" list which is actually two singly linked lists. One "traverses" this list by popping nodes off one and pushing them onto the other. The user has easy access to the heads of both lists. You could probably do the same with trees but this is a pretty unpleasant pattern.