Why did President Lyndon Johnson expand America's role in the war in 1965? How and why did President Richard Nixon change America’s role in this conflict?

After the USS Maddox was attacked in the Gulf of Tonkin, Lyndon Johnson increased the number of ground troops in Vietnam with the Gulf of Tonkin Resolution in 1965. He did this in order to stop Vietcong incursions into South Vietnam. He believed that the war could be won with enough American firepower and manpower. He did not recognize the resilience of the North Vietnamese government and their desire to unite Vietnam under one flag. Johnson thought that a show of strength in Vietnam would show to the US's allies that America was serious about containing communism. Johnson was operating under belief in the domino theory, which stated that if one country switched to communism then others would soon follow. Johnson knew that some in Washington never forgave Harry Truman for "letting" China go to the Communists, and he did not want to make that mistake in South Vietnam.
Nixon took a different approach to solving the Vietnam War.  Nixon was elected under the promise of getting the United States out of Vietnam "with honor." The war-weary American people turned out the Democrats in 1968 with the intention of withdrawing from Vietnam with some sort of victory. Nixon used special operations soldiers to invade the Ho Chi Minh Trail in Laos and Cambodia. He also insisted that the South Vietnamese fight more of their own battles through a process called Vietnamization. Nixon also increased bombing campaigns in North Vietnam in order to bring the Hanoi government to the peace table. This led to the Paris Peace Accords and the last American troops leaving Vietnam in 1973.