How do I create an instance of a composite templated upon a different type?

I am creating composite objects from various building blocks. This is best
described with an example.

template <class T>
class Behavior
{
public:

    virtual void DoSomething(T& t) = 0;

    virtual void AddChild(Behavior<T>* child){}
};

//------------------------------------------------------------------

template <class T>
class CompositeBehavior : public Behavior<T>
{
    list<Behavior<T> > children;

public:

    virtual void DoSomething(T& t) = 0;

    virtual void AddChild(Behavior<T>* child){children.push_back(child);}
};

//-------------------------------------------------------------------

template <class T>
class SelectBehavior : public Composite<T>
{
    void DoSomething(T& t) ;
};

//-------------------------------------------------------------------

template <class T>
class ConcreteBehavior : public Behavior<T>
{
    void DoSomething(T& t) ;
};

So I create a composite out of these building blocks.

Behavior<Dog>* pDogRoot = new SelectBehavior<Dog>();
pDogRoot ->AddChild(new ConcreteBehavior<Dog>());
pDogRoot ->AddChild(new ConcreteBehavior<Dog>());

I'd like to be able to create an instance of this composite as though it was
created using a different type, say a Cat, but without going through the
process of creating it all over again by hand (the composites can get
complex). Ideally I'd like to be able to pass some function pRoot and it
return an instance created for the specified type.

Behavior<Cat>* pCatRoot = MakeInstance<Cat>(pDogRoot);

How could i go about doing this? What design patterns might be relevent? (is
it even possible?)

Many thanks for any help.