The justification for celebrating Palm Sunday is found in the gospel accounts. Some denominations teach that, early on a Sunday morning, Jesus left Bethany and went to Bethphage. In Bethphage he mounts a colt and journeys to Jerusalem, which fulfills Biblical prophecy (Zechariah 9:9). As he travels towards the city, a large crowd of people begin to gather and cry out praises to God. They place cut palm tree branches, and even their clothes, in front of the colt in honor of the great King (Matthew 21:1 - 9, John 12:12 -15).
The celebratory use of palms is also found in Scripture regarding the keeping of God's Feast Days known as the Feast of Tabernacles.
39. Also in the fifteenth day of the seventh month, when you have gathered in the fruit of the land, you shall keep a feast to the LORD seven days . . .
40. And you shall take the boughs of beautiful trees for yourselves on the first day, branches of palm trees, and the boughs of thick trees, and willows of the brook. And you shall rejoice before the LORD your God . . . (Leviticus 23:39 - 40, HBFV).
According to Adam Clarke's Commentary on the Bible, these and other tree branches were emblematical of victory and success. In modern churches who celebrate Palm Sunday, palms (or parts of yew, willow, or other native trees) may be distributed to worshippers. These tree parts, in some cases, are also burned, blessed, and have their ashes kept for the observance of Ash Wednesday.
The Biblical truth is that Jesus did not mount a colt and enter Jerusalem, for the last time, on a Sunday. Our timeline shows that Christ entered Jerusalem on a Thursday in 30 A.D. He would ultimately suffer and die for the sins of the world not on Good Friday but on Wednesday.