When tree will go out of scope, their nodes memory will be reclaimed automatically ("dropping" in Rust terms) by automatic garbage collector built-in into compiler: `drop(root)->drop(leaf_a),drop(leaf_b)`. However, if leaf will have reference to its parent, then an infinite loop occurs: `drop(root)->drop(leaf_a)->drop(root)->drop(leaf_a)...`
The solution is to use an alternative automatic garbage collector instead of compiler built-in: arenas, RC with weak references, or Mark&Sweep GC. Arenas have better performance, because they drop all nodes at once. The easiest way to quickly implement an arena in safe Rust is to use vector (array) of nodes and used indices instead of direct references.
You've said it would be obviously wrong "even in a language with garbage collector, such as JavaScript." But, obviously, most modern GC can handle cyclic references just fine.
But even languages with Tricolour Mark&Sweep GC, which handles cyclic references just fine, it's still possible to make memory leak via a dangling reference to a node in a complex cross-linked graph, because language and GC allows that.
Rust by default forbids cyclic references, by forcing to use trees, which completely avoids the problem.
Rust uses double linked lists, they are not harder to implement in Rust than in any other language. Moreover, built-in borrow checker will help to implement them properly, without memory leaks or use-after-free. What is your point?
The solution is to use an alternative automatic garbage collector instead of compiler built-in: arenas, RC with weak references, or Mark&Sweep GC. Arenas have better performance, because they drop all nodes at once. The easiest way to quickly implement an arena in safe Rust is to use vector (array) of nodes and used indices instead of direct references.