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C# operator overloading: Object.Equals(object o) & Object.GetHashCode()

So I am creating a BST and want it to be a generic tree, right now I am developing the Node<T> class. I need some help with some of the operator overloading specifics. Class is below:

public class Node<T> where T : IComparable
{
    //Private member data
    private T data;
    private Node<T> left;
    private Node<T> right;
    //private readonly IComparer<T> _comparer;

    //Node constructor
    public Node()
    {
        data = default(T); //
        left = null;
        right = null;
    }

    //Setters/getters for node private data members
    public T Data
    {
        get { return data; }
        set { data = value; }
    }

    public Node<T> Left
    {
        get { return left; }
        set { left = value; }
    }

    public Node<T> Right
    {
        get { return right; }
        set { right = value; }
    }

    public static bool operator ==(Node<T> lhs, Node<T> rhs)
    {
        if((lhs.Data).CompareTo(rhs.Data) == 0)
        {
            return true;
        }
        else
        {
            return false;
        }

    }

    public static bool operator !=(Node<T> lhs, Node<T> rhs)
    {
        if (lhs.Data.CompareTo(rhs.Data) != 0)
        {
            return true;
        }
        else
        {
            return false;
        }
    }
}

But Visual Studio (and sources I have seen online) say I need to overload the Object.Equals(object o) method and also the Object.GetHashCode.

From what I know about .Equals just through using C#, it's like value type semantics, or will compare the values of 2 objects, instead of their references, aka checking if they are actually the same object, which I think is what == usually does when comparing objects. So this is what I tried to do:

public override bool Equals(Object obj)
{
    if ((lhs.Data).CompareTo(rhs.Data) == 0)
    {
        return true;
    }
    else
    {
        return false;
    }
}

It's basically the same as my == operator, but it doesn't work because the lhs/rhs parameters don't exist. I have no idea how do accomplish this for my case since how I believe it will be called is n1.Equals(n2) and will check to see if the data values are the same for the nodes. Examples online are unclear to me. I also have no idea why I have to involve this hash method at all, but I am still trying to do research on that. Mostly curious about the Equals override for now.

Make sure you implement the Equals method according to the MS guide. Here is my snippet. I use all the time:

        // override object.Equals
public override bool Equals(object obj)
    {
        // shortcut
        if (object.ReferenceEquals(this, obj))
        {
            return true;
        }

        // check for null and make sure we do not break oop / inheritance
        if (obj == null || GetType() != obj.GetType())
        {
            return false;
        }

        // TODO: write your implementation of Equals() here
        // do not call base.equals !        
        throw new NotImplementedException();

        return false;
    }

    public static bool operator ==(Class lobj, Class robj)
    {
        // users expect == working the same way as Equals
        return object.Equals(lobj, robj);
    }

    public static bool operator !=(Class lobj, Class robj)
    {
        return !object.Equals(lobj, robj);
    }

    // override object.GetHashCode
    public override int GetHashCode()
    {
        // TODO: write your implementation of GetHashCode() here
        // return field.GetHashCode() ^ field2.GetHashCode() ^ base.GetHashCode();
        // or simply return the unique id if any

        throw new NotImplementedException();
    }

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