how to compare and match two generic types

Short answer, without knowing anything about the type, you can't do any form of sorting or advanced value checking, because it would make no sense. (What if someone made a YourTree<double>? Or a YourTree<Socket>? Then looking for the first few characters would be meaningless.)

The exceptions are things that work on Object, so you can:
- group them together by value equality using Object.Equals or EqualityComparer as Kim says
- use their hashcode, though this is generally not useful
- use their string representation (ToString()), which might be what you want

It sounds like you want to restrict your T to classes which have particular operations on them, namely, providing a name, and possibly be comparable. That means you need an interface, for example

public interface ITreeNode : IComparable<ITreeNode> {
 string Name { get; }
}

... and then you need to restrict the T on your main class to support that interface:

public class MyBinaryTree<T> where T: ITreeNode {
...
}

Now, you can't create a MyBinaryTree<int> (int is not an ITreeNode), so you need to create ITreeNode subclasses for the data types you want to be in the tree. You can have a default one that looks like

public class TreeNode<T> : ITreeNode {
 T inner;

 public TreeNode(T inner) { this.inner = inner; }

 public int CompareTo(T other){
  if(inner is IComparable<T>) return ((IComparable<T>)inner).CompareTo(IComparable<T>)other);
  else return StringComparer.CurrentCulture.Compare(inner, other);
 }

 public string Name { get { return inner.Name; } }
}

... which wraps any type and uses its comparer if it has one, otherwise the string representation. (I think the native numeric types support IComparable<T> so they should sort correctly.) Now you can do

var myTree = new MyBinaryTree<TreeNode<int>>();

.. and within MyBinaryTree, you can use the methods of ITreeNode (i.e. the name), and also treat the nodes as IComparable<T> (i.e. pass them to sorting routines).

Hopefully that gives you enough information that you can work out exactly how to apply it to your situation.

E: Naturally, this was made up on the spot, not tested, and probably won't even compile. Also, generic types and HTML don't really play well together.