Covert the given AST into a doubly linked tree while assigning ids (so it stays serializable).
The original decoration of the ast nodes (probably is nothing as the id decoration is most likely the first step to be performed after extraction)
The root of the AST to convert
A decorated AST based on the input and the id provider.
Covert the given AST into a doubly linked tree while assigning ids (so it stays serializable).