How to convert an expression tree to a partial SQL query?

Yes it is possible, you can parse a LINQ expression tree using the visitor pattern. You would need to construct a query translator by subclassing ExpressionVisitor like below. By hooking into the correct points you can use the translator to construct your SQL string from your LINQ expression. Note that the code below only deals with basic where/orderby/skip/take clauses, but you can fill it out with more as needed. Hopefully it serves as a good first step.

public class MyQueryTranslator : ExpressionVisitor
{
    private StringBuilder sb;
    private string _orderBy = string.Empty;
    private int? _skip = null;
    private int? _take = null;
    private string _whereClause = string.Empty;

    public int? Skip
    {
        get
        {
            return _skip;
        }
    }

    public int? Take
    {
        get
        {
            return _take;
        }
    }

    public string OrderBy
    {
        get
        {
            return _orderBy;
        }
    }

    public string WhereClause
    {
        get
        {
            return _whereClause;
        }
    }

    public MyQueryTranslator()
    {
    }

    public string Translate(Expression expression)
    {
        this.sb = new StringBuilder();
        this.Visit(expression);
        _whereClause = this.sb.ToString();
        return _whereClause;
    }

    private static Expression StripQuotes(Expression e)
    {
        while (e.NodeType == ExpressionType.Quote)
        {
            e = ((UnaryExpression)e).Operand;
        }
        return e;
    }

    protected override Expression VisitMethodCall(MethodCallExpression m)
    {
        if (m.Method.DeclaringType == typeof(Queryable) && m.Method.Name == "Where")
        {
            this.Visit(m.Arguments[0]);
            LambdaExpression lambda = (LambdaExpression)StripQuotes(m.Arguments[1]);
            this.Visit(lambda.Body);
            return m;
        }
        else if (m.Method.Name == "Take")
        {
            if (this.ParseTakeExpression(m))
            {
                Expression nextExpression = m.Arguments[0];
                return this.Visit(nextExpression);
            }
        }
        else if (m.Method.Name == "Skip")
        {
            if (this.ParseSkipExpression(m))
            {
                Expression nextExpression = m.Arguments[0];
                return this.Visit(nextExpression);
            }
        }
        else if (m.Method.Name == "OrderBy")
        {
            if (this.ParseOrderByExpression(m, "ASC"))
            {
                Expression nextExpression = m.Arguments[0];
                return this.Visit(nextExpression);
            }
        }
        else if (m.Method.Name == "OrderByDescending")
        {
            if (this.ParseOrderByExpression(m, "DESC"))
            {
                Expression nextExpression = m.Arguments[0];
                return this.Visit(nextExpression);
            }
        }

        throw new NotSupportedException(string.Format("The method '{0}' is not supported", m.Method.Name));
    }

    protected override Expression VisitUnary(UnaryExpression u)
    {
        switch (u.NodeType)
        {
            case ExpressionType.Not:
                sb.Append(" NOT ");
                this.Visit(u.Operand);
                break;
            case ExpressionType.Convert:
                this.Visit(u.Operand);
                break;
            default:
                throw new NotSupportedException(string.Format("The unary operator '{0}' is not supported", u.NodeType));
        }
        return u;
    }


    /// <summary>
    /// 
    /// </summary>
    /// <param name="b"></param>
    /// <returns></returns>
    protected override Expression VisitBinary(BinaryExpression b)
    {
        sb.Append("(");
        this.Visit(b.Left);

        switch (b.NodeType)
        {
            case ExpressionType.And:
                sb.Append(" AND ");
                break;

            case ExpressionType.AndAlso:
                sb.Append(" AND ");
                break;

            case ExpressionType.Or:
                sb.Append(" OR ");
                break;

            case ExpressionType.OrElse:
                sb.Append(" OR ");
                break;

            case ExpressionType.Equal:
                if (IsNullConstant(b.Right))
                {
                    sb.Append(" IS ");
                }
                else
                {
                    sb.Append(" = ");
                }
                break;

            case ExpressionType.NotEqual:
                if (IsNullConstant(b.Right))
                {
                    sb.Append(" IS NOT ");
                }
                else
                {
                    sb.Append(" <> ");
                }
                break;

            case ExpressionType.LessThan:
                sb.Append(" < ");
                break;

            case ExpressionType.LessThanOrEqual:
                sb.Append(" <= ");
                break;

            case ExpressionType.GreaterThan:
                sb.Append(" > ");
                break;

            case ExpressionType.GreaterThanOrEqual:
                sb.Append(" >= ");
                break;

            default:
                throw new NotSupportedException(string.Format("The binary operator '{0}' is not supported", b.NodeType));

        }

        this.Visit(b.Right);
        sb.Append(")");
        return b;
    }

    protected override Expression VisitConstant(ConstantExpression c)
    {
        IQueryable q = c.Value as IQueryable;

        if (q == null && c.Value == null)
        {
            sb.Append("NULL");
        }
        else if (q == null)
        {
            switch (Type.GetTypeCode(c.Value.GetType()))
            {
                case TypeCode.Boolean:
                    sb.Append(((bool)c.Value) ? 1 : 0);
                    break;

                case TypeCode.String:
                    sb.Append("'");
                    sb.Append(c.Value);
                    sb.Append("'");
                    break;

                case TypeCode.DateTime:
                    sb.Append("'");
                    sb.Append(c.Value);
                    sb.Append("'");
                    break;

                case TypeCode.Object:
                    throw new NotSupportedException(string.Format("The constant for '{0}' is not supported", c.Value));

                default:
                    sb.Append(c.Value);
                    break;
            }
        }

        return c;
    }

    protected override Expression VisitMember(MemberExpression m)
    {
        if (m.Expression != null && m.Expression.NodeType == ExpressionType.Parameter)
        {
            sb.Append(m.Member.Name);
            return m;
        }

        throw new NotSupportedException(string.Format("The member '{0}' is not supported", m.Member.Name));
    }

    protected bool IsNullConstant(Expression exp)
    {
        return (exp.NodeType == ExpressionType.Constant && ((ConstantExpression)exp).Value == null);
    }

    private bool ParseOrderByExpression(MethodCallExpression expression, string order)
    {
        UnaryExpression unary = (UnaryExpression)expression.Arguments[1];
        LambdaExpression lambdaExpression = (LambdaExpression)unary.Operand;

        lambdaExpression = (LambdaExpression)Evaluator.PartialEval(lambdaExpression);

        MemberExpression body = lambdaExpression.Body as MemberExpression;
        if (body != null)
        {
            if (string.IsNullOrEmpty(_orderBy))
            {
                _orderBy = string.Format("{0} {1}", body.Member.Name, order);
            }
            else
            {
                _orderBy = string.Format("{0}, {1} {2}", _orderBy, body.Member.Name, order);
            }

            return true;
        }

        return false;
    }

    private bool ParseTakeExpression(MethodCallExpression expression)
    {
        ConstantExpression sizeExpression = (ConstantExpression)expression.Arguments[1];

        int size;
        if (int.TryParse(sizeExpression.Value.ToString(), out size))
        {
            _take = size;
            return true;
        }

        return false;
    }

    private bool ParseSkipExpression(MethodCallExpression expression)
    {
        ConstantExpression sizeExpression = (ConstantExpression)expression.Arguments[1];

        int size;
        if (int.TryParse(sizeExpression.Value.ToString(), out size))
        {
            _skip = size;
            return true;
        }

        return false;
    }
}

Then visit the expression by calling:

var translator = new MyQueryTranslator();
string whereClause = translator.Translate(expression);

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