A little summary:
As people have already noted, there are two different lines in the evolutionary table of Yaks. There are the "light" Yaks, and then there are the "heavy" Yaks. Therefore, people tend to get confused if they think of the designation methods they are familiar with. The final two versions of the two Yak types are the Yak-3 and the Yak-9U. The earliest of the two Yak types are Yak-1 and Yak-7.
The prototype for all Yaks was the plane designated "I-26". After the I-26 production types became the Yak-1, it evolved into Yak-1B and Yak-3. The VVS was initially very pleased with the I-26, and it was therefore also produced astwo-seat trainer version "UTI-26" (Uchbeno Trenirovochny Istrebitel) in March 4th of 1940. The UTI-26, after going through two versions(UTI-26-1, UTI-26-2), was redesignated and modified into Yak-7UTI trainer.
After "Barbarossa", the VVS thought up a way to deal with plane shortages by modifying existing two-seat trainers into single-seat fighters, and thus the Yak-7UTI, with the rear trainer seat removed, became the Yak-7. The Yak-7 was further developed into the Yak-7A and the Yak-7B, and after 1942, it evolved into the Yak-7DI, which can be considered as an "early Yak-9". Thus, the mainstay of VVS fighters, the Yak-9, evolved from the UTI-26, and reached its final stage with the coming of the Yak-9U.
Meanwhile, the Yak-1, which was considered the only fighter potent enough to match what the Germans put up, was also going through various changes. The mechanics of the 42IAP led by Major F. I. Shinkarenko had come up with some field modifications based on the pilots' experiences and complaints. The rear fuselage was lowered, and the canopy was changed to "bubble" types. These field-mod Yak-1s were known as "Shinkarenko's Yaks", and became a direct base for the improved version - Yak-1B.
With the M-105PF engine and armed with 1xUBS machine gun and 1xShVAK 20mm cannon, the Yak-1B was the most formidabble opponent of the German Bf109F-4 and even a worthy match for the latest German fighter Bf109G-2 at low altitudes: a very maneuverable fighter, doing 321mph at sea-level, 363mph at approximately 12k feet, 5.6 minutes to 15k feet, 18 seconds to 360 degrees turn. Armed with the best of WWII heavy machine guns 12.7mm UBSmg x1 (200 rpg), and 20mm ShVAKcan x1 (140rpg).
After 1942, the Yak-1B was further experimented with improved engines and armament, which produced the prototype Yak-1M, which was to become the Yak-3.
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Some specs:
* Yak-7UTI(1941)
M-105PA(1,100hp)
360mph @ 13,500ft
18 seconds to 360d turn
6.6 minutes to 15,000ft
* Yak-7(1941)
M-105PA(1,100hp)
344mph @ 13,500ft
24 seconds to 360d turn
6.8 minutes to 15,000ft
* Yak-7B(1942)
M-105PA(1,100hp)
356mph @ 15,000ft
21 seconds to 360d turn
6.5 minutes to 15,000ft
* Yak-1B(1942)
M-105PF(1,180hp)
363mph @ 11,550ft
18 seconds to 360d turn
5.6 minutes to 15,000ft
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We really need some of these early VVS Yaks

I-16 and LaGG-3 for pre-1941, Yak-7 for 1941, and Yak-1B for 1942. Yak-9 for 1943. Yak-3 for 1943~1944. These six planes + one or two more VVS bombers, with the four previously existing VVS fighters, would finish the VVS fighter plane set with about twelve planes - almost the same as US and LW.
The VVS needs all of those planes. Soviet Russia was literally the largest participant of WWII in almost all aspects.