A friend recently asked me: are there any tricycle-gear biplanes? After hunting around, we found the Waco AVN-8, but that's all. This causes me to ask: is there an aerodynamic or other design consideration which makes a tricycle gear a bad choice for a biplane, or is it just an evolutionary coincidence that the move to single-wing designs shortly preceded the move to tricycle gear?
It is a bit more than a coincidence. You might also ask why there are so few biplanes with retractable landing gear. The reason is similar.
- Biplanes are propeller-driven (except for this one), and in almost all cases the propeller is in the front. Efficient propellers are large and require a longer landing gear when a tricycle gear is used. Granted, the plane will lift its tail during the ground run so the prop comes down a bit, but that happens over a flat and clean area, so lower tolerances can be afforded during the ground run than during all other ground operations.
- Tricycle gears produce substantially more drag. The nose gear sits directly behind the propeller and in its slipstream, so its drag contribution is typically 40% of the whole gear drag. Tricycle gears became less of a drag (pun intended) when gear retraction became standard.
- Biplanes are built light and the heavy engine cannot have much distance to the center of gravity. Where do you mount the nose gear? There is preciously little room for another piece of structure. And once that nosegear is crafted on the plane, its distance from the main gear is rather small. Any bump on the field will produce strong pitch oscillations when hit by the nose wheel. Better to have a tailwheel or skid at some distance from the main wheels.
- Tricycle gears became a necessity with higher wing loadings and higher landing speeds. Biplanes don't need powerful brakes. On the contrary, on a grass field a taildragger is easier to handle. And with their short ground runs, biplanes often operate from grass fields.
Not that tricycle gears weren't tried. See the picture of a Breguet 6 below (source), conveniently proving that pusher biplanes were also designed. Gun synchronisation removed the last reason not to place the propeller in front.
The biplane and tailwheel configurations are not directly related, but are both symptoms of the historical quest for light weight.
Given a rather inefficient engine in the 70-150 hp class, lifting both pilot and any kind of payload is a challenge. All-up weight must be severely limited. The biplane is more structurally efficient or, to put it the other way, gives such an aircraft better performance, than a monoplane. Similarly, a tail wheel is simply smaller and lighter than a nose wheel.
200-2,000 hp engines brought the monoplane speeds which the biplane could not reach and so the biplane was all but abandoned. But the need for light weight remained almost as pressing and, with the bulky high-power engine also occupying the space at the nose and pilots already used to tail wheels, the tail wheel remained customary.
But these fast monoplanes had high takeoff and landing speeds and, consequently, high ground speeds before the rudder could take effect. Engine torque did not help matters. As engine powers and takeoff speeds increased rapidly, many a WWII fighter would ground-loop while taxiing or landing and sometimes kill it pilot in the process. The nose wheel was found to be more manageable and by the end of WWII, nose wheels were beginning to appear.
By coincidence, the 1940s also marked the arrival of the jet engine. The prototype Messerschmitt Me 262 had a tailwheel. However the ground handling of those early but unprecedentedly fast Me 262s was nerve-wracking, while suddenly the nose had empty space and a consequent need for noseweight, and engines were becoming so powerful that weight was less of a problem. So Messerschmitt changed over to a nose wheel for the production version. Most of the jets developed during WWII had nose wheels from Day One.
The propeller position, mentioned in some answers, is not especially relevant to biplanes. It is just coincidence that most piston-engines planes, bi- or mono-, are tractors. Pusher biplanes such as the RAF FE2 and Vickers FE8 also had tail wheels as standard (with some also having skids or small nose wheels mounted off the ground on the main u/c structure, in order to prevent nose-over). It was only during WWII that the pusher monoplane was found to be handicapped by the tail wheel and these designs adopted the nose wheel whose star was fortuitously rising. The tail wheel even outlasted the WWII monoplane era and in some cases persisted right through to the postwar jet age. For example Supermarine took another decade to wake up and smell the coffee, as they say, with the swept-wing Supermarine Type 510 and its successors appearing through the late 1940s and even as late as 1950.
As you suspected, this in an evolutionary coincidence. As construction methods and materials improved, multiple stacked wings became obsolete being less efficient to otherwise comparable single wing design.
When biplanes were extinct, by chance the tricycle gear arrangement started its "triumph". These two are separate advancements in engineering. One could very well combine biplane wing arrangement with tricycle gear, it's just that there's no sense in using a biplane arrangement anymore.
There is another reason, sort of a lost art among modern aircraft designers: to balance vertical and lateral drag.
Early aircraft flew at much slower airspeeds, making them far more susceptible to cross wind gusts. Large forward set wheels underneath and in front of the fuselage helped balance side wind drag forces on the vertical tail, while cleverly giving the rear vertical airfoil the advantage at lower (yaw) angles of attack.
The biplane configuration also reduces side wind gust roll torque compared with monoplanes of equal wing area. (Monoplanes solve this with the "high wing" configuration, also seen in paper airplanes).
So, fixed forward raked landing gear really serve an aerodynamic purpose as well (the Fokker DR-1 even put a fourth wing there too). The ultimate advantage of tricycle gear could be seen with the high speed Space Shuttle landing: to allow reduction of wing AOA to a non lifting condition while rolling out at high speed. Higher approach and landing speeds make an aircraft less susceptible to cross wind effects, making this landing technique preferable (in all sorts of weather) provided there is adequate runway length.
One may wonder if the biplane could make a comeback as a recreational ultralight. One might feel safer with that design as compared with others.