Converting between infix, prefix and postfix notation?

Question

Algorithms

As far as I understand, the transition from infix to prefix or postfix notation is easier to do using the algorithm proposed by Dijkstrom (using the stack), and other transformations are through building an abstract syntax tree (parse tree, expression parse tree) and traversing it in various ways.
I'm wondering if there are implementations, pseudocode or descriptions of the algorithms PrefixToInfix, PostfixToInfix, PrefixToPostfix, PostfixToPrefix that are not based on expression trees , or if there are ways to recursively parse an infix form with incomplete brackets and with non-single-letter variables, build a parse tree from it .
The plan is to implement a program that receives an expression in one of the notations as input, and an expression in any other notation or a parse tree as output.