In Country
    Just because a unit was deployed does not mean it served full strength. This would've been truly rare. In addition the following units arrived in country most often in parts, slowly drizziling in throughout the year.

Unit                                                                                                Honorifics
1965
1st Cav Division(Airmobile)                                                             The First Team
3rd Marine Division                                                                         Fidelity, Honor, Valor
1st Infantry Division(Mechanized)                                                    Big Red One
173d Airborne Brigade                                                                    Sky Soldiers/The Herd
1st, Brigade 101st Airborne Division                                                Screaming Eagles\Battered Bastards of Bastagone
1966
4th Infantry Division                                                                         Ivy Division
9th Infantry Division                                                                        Old Reliables
25th Infantry Division(Light)                                                            Tropical Lighting
1st Marine Division
11th Armored Cav Regiment                                                           The Black Horse Regiment
196th    Light Infantry Brigade(LIB)                                                 Chargers/Ahead Of The Rest
199th LIB                                                                                     Redcatcher
1967
101st Airborne Division                                                                  Screaming Eagles\Battered Bastards of Bastagone
23rd Infantry Division                                                                     Americal Division
198th  LIB                                                                                     Bold and Brave
1968
82nd Airborne                                                                               All American

Notes
101st Airborne Division: Fought at Hamburger Hill and in the re-capture of Hue during Tet.
173d Airborne: Was the first US ground force to be sent to Vietnam. Worked with the Australians a lot in the early days. During Operation Junction City, they preformed the wars only combat drop.
82nd Airborne: Served in the Dominican in 1965. After returning it served as a strategic reserve before being sent to Vietnam.
9th Infantry Division:  The 2nd brigade help make up the Mobile Riverine Force in the Mekong Delta.
The Americal Division: Was re-formed out of the 11th LIB, 196th LIB, and the 198th LIB. Here's the words from the Divisions official history:
On September 22, 1967, Brigadier General Samuel W. Koster assumed command of the task
  force, replacing Major General Richard T. Knowles, and three days later Task Force Oregon
  became the Americal Division, composed of the 196th, 198th, and the 11th Light Infantry Brigades,
  even though the latter two organizations were still training in the United States.

Sources: Vietnam Order of Battle, Changing Orders, and various webpages

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