The answer is: "Yes, at least 99% of the time."
The iPad can use almost any iPhone or iPod Touch application, including "Traces of the Past," a genealogy program designed for the iPhone and iPod Touch. However, the iPad has a large screen whereas the iPhone and iPod Touch have smaller screens. As a result, the iPad will display the information as a small display in the center of the iPad's screen with black surrounds (or at an ugly 2x scaled).
For instance, here is a picture of an iPhone application running on the iPad:
While tough to see in this small screen capture, the 2x display will be slightly grainy as everything is "blown up" to fill the screen.
In contrast, here is the same application after it has been re-written specifically for the iPad's larger screen:
Of course, all this assumes that the iPad also contains the hardware required by the application. For instance, my iPad does not contain a GPS. All iPhones manufactured in the past two years do contain a GPS. Therefore, any iPhone application that requires GPS will work on newer iPhones, but not on the iPads that do not have a GPS installed. (The more expensive 3G-equipped iPads DO have GPS.) However, hardware compatibility is not an issue for any of the iPhone or iPad Touch genealogy programs available today. They all will work on today's iPad, although at a small size with black surrounds or as a 2x display.
For reference, here is the iPhone:
Here is the (larger) iPad: