Why do Birds have Tails? The Tail as a Drag Reducing Flap, and Trim Control

Birds have tails, bats do not. Does this fundamental difference in flight morphology reveal a difference in flight capability, and if so are birds or bats better fliers? I use Munkʹs stagger theorem, and Prandtlʹs relation for the induced drag of a biplane to show that for a given lift, and given wingspan, the induced drag of the wing-tail combination is lower that the induced drag of a wing alone. However the same reduction in induced drag could be achieved by slightly increasing the wingspan. While increasing the wingspan reduces induced drag, it can also increase profile drag and wing inertia. Induced drag is dominant at low speeds and during turns. Profile drag dominates at high speeds. The tail allows birds to have the wings needed for efficient cruising and high speed flight (when the tail can be furled giving little drag), at the same time the tail can be spread at low speeds or during turns to reduce induced drag. The tail can play a role in maintaining stability and balance, and it appears that the stability of birds is tailored so that the tail is required to generate lift at low speeds, when the interaction between the wings and the tail can also most effectively reduce induced drag.