binary tree dictionary using c #

This is how the binary tree is designed: http://en.wikipedia.org/wiki/Binary_tree[^].

This is how it is used for binary search: http://en.wikipedia.org/wiki/Binary_search_tree[^].

If you look at the algorithms, you will see that keys need a comparison criterion, but only "less then" relationship is actually used. You need to use the set of values that allows for the ordering: http://en.wikipedia.org/wiki/Total_order[^].

You need to define the order relationship for the the key values. To make is generic, you can define the interface for ordering (there is no a big need to define the '<' operator, but you can do it if it is convenient for your). Or you can use the more complex and general interface which is already available: System.IComparable:
http://msdn.microsoft.com/en-us/library/system.icomparable.aspx[^].

Again, you can derive from this interface, if you want a comparison operator. Your key type should implement this interface, and then you can define a generic your node type assuming that the key type is IComparable, which is done using the generic type constraint:
http://msdn.microsoft.com/en-us/library/bb384067.aspx[^].

For a tree node, use generic class would look like this:

class Node<KEY_TYPE, VALUE_TYPE> : IComparable where KEY_TYPE : System.IComparable {

    KEY_TYPE key; //
    VALUE_TYPE data; // anything you need to put in data
    Node<KEY_TYPE, VALUE_TYPE> left, right;

    int IComparable.CompareTo(object @object) {
        Node<KEY_TYPE, VALUE_TYPE> nodeObject = @object as Node<KEY_TYPE, VALUE_TYPE>; // use it for comparison with "this"
        // implement comparison between @object and this
        // take into account that @object can be null
        // and that @object can be of different type
    }

    // implement constructors, operations with left, right, etc...

}

Note that I suggested to implement IComparable also for the Node type. This way, you won't need to compare keys, will be able to compare nodes, immediately.

This will give you a cultured and convenient way to implement tree building algorithm and search.

The interface of your dictionary should identical to the one of the class System.Collections.Generic.Dictionary<>:
http://msdn.microsoft.com/en-us/library/xfhwa508%28v=vs.110%29.aspx[^].

There is nothing to invent, use it one-to-one. The algorithms are described in the second link. Good luck.

[EDIT]

In response to a good note by Matt T Heffron:

Asymptotically, this algorithm is not the fastest for big volumes of data; it provides computational time complexity of O(ln N), while System.Collections.Generic.Dictionary (and two more classes based on bucket storage and hash function) has time complexity of O(1).

Please see:
http://en.wikipedia.org/wiki/Big_O_notation[^],
http://en.wikipedia.org/wiki/Time_complexity[^],
http://en.wikipedia.org/wiki/Asymptotic_complexity[^],
http://en.wikipedia.org/wiki/Asymptote[^].

—SA