The 357 and Trooper linage is as follows......
Colt introduced the original Trooper and 357 in 1953.
The 357 was Colt's first "I" frame model that had a firing pin in the frame, and was the worlds first medium frame double action revolver in .357 Magnum.
Prior to this, all .357 Magnums were built on large frames.
Colt used the New Service and S&W used their large "N" frame.
The Trooper was an "E" frame model with the firing pin mounted on the hammer.
The Trooper was available in .38 Special and in .22LR. The .22 was only available with a 4 inch barrel.
Both the 357 and Trooper were available with any combination of Service hammer, Target hammer, Service grips, or Target grips.
With the introduction of the super-premium Python in 1955 sales of the 357 began to drop, so in 1961 Colt discontinued the 357 AND the "E" frame Trooper and began selling the "I" frame under the Trooper name in .38 Special or .357 Magnum.
The .22LR model was still cataloged until 1969 but it's thought that actual production was ended around 1960.
In 1969 Colt discontinued most of the old models and introduced a totally new design, the "J" frame Mark III.
This was the first modern transfer bar safety-ignition action with nothing in common with the old models.
In 1983 Colt did an upgrade to the Mark III series and brought out a "V" frame Trooper Mark V.
In 1985 Colt fitted the Mark V Trooper with a different profile barrel and first made it in stainless steel as the King Cobra.
As is often the case with Colt, there's name confusion between the original Troopers and the new Trooper Mark III and Mark V.
People often call the Trooper Mark III simply a Trooper but as above they're totally different models.