How do I infer the usage of parentheses when translating an expression tree?

Question

I am working on translating an expression tree to a format that resembles infix notation; I am not evaluating the tree or executing its operations. The tree contains both logical and relational operations, and I would like to emit parentheses in an intelligent manner during the translation.

To illustrate, consider the following contrived expression:

a < x & (a < y | a == c) & a != d

If I walk the expression tree produced by this expression in-order, then I will print out the following expression, which is incorrect.

a < x & a < y | a == c & a != d
// equivalent to (a < x & a < y) | (a == c & a != d)

Alternatively, I can again perform an in-order traversal but emit parentheses before and after processing a binary expression. This will produce the following correct expression, but with several redundant parentheses.

(((a < x) & ((a < y) | (a == c))) & (a != d))

Is there an expression tree traversal algorithm that produces optimally-parenthesized expressions?

For reference, here is a snippet of the ExpressionVisitor I am using to inspect the tree.

class MyVisitor : ExpressionVisitor
{
    protected override Expression VisitBinary(BinaryExpression node)
    {
        Console.Write("(");

        Visit(node.Left);
        Console.WriteLine(node.NodeType.ToString());
        Visit(node.Right);

        Console.Write(")");

        return node;
    }

    // VisitConstant, VisitMember, and VisitParameter omitted for brevity.
}

Answer 1

I've accepted the answer of Dialecticus as it provides a good basis for implementing this algorithm. The only issue with this answer is that it requires that the VisitBinary() method know about its parent caller as a method argument, which is not feasible since these methods are overloads of a base method.

I provide the following solution, which uses a similar algorithm, but applies the check to emit parentheses in the parent call for the child nodes of the expression tree.

class MyVisitor : ExpressionVisitor
{
    private readonly IComparer<ExpressionType> m_comparer = new OperatorPrecedenceComparer();

    protected override Expression VisitBinary(BinaryExpression node)
    {
        Visit(node, node.Left);
        Console.Write(node.NodeType.ToString());
        Visit(node, node.Right);

        return node;
    }

    private void Visit(Expression parent, Expression child)
    {
        if (m_comparer.Compare(child.NodeType, parent.NodeType) < 0)
        {
            Console.Write("(");
            base.Visit(child);
            Console.Write(")");
        }
        else
        {
            base.Visit(child);
        }
    }

    // VisitConstant, VisitMember, and VisitParameter omitted for brevity.
}

The precedence comparison function is implemented as an IComparer<ExpressionType>, which applies the C# rules of operator precedence.

class OperatorPrecedenceComparer : Comparer<ExpressionType>
{
    public override int Compare(ExpressionType x, ExpressionType y)
    {
        return Precedence(x).CompareTo(Precedence(y));
    }

    private int Precedence(ExpressionType expressionType)
    {
        switch(expressionType) { /* group expressions and return precedence ordinal * }
    }
}