The Jesse Trees comes from very ancient Christmas Culture and it was first started in medieval times. They are used to help tell the story of the Bible from creation to the Christmas Story. The name comes from Jesse who was the Father of the great Jewish King David. One prophecy in the Bible, in the book of Isaiah, says many things about the Jesse trees read the following to know more:

1 A shoot will come up from the stump of Jesse; from his roots, a Branch will bear fruit. 
2 The Spirit of the LORD will rest on him-- the Spirit of wisdom and of understanding, the Spirit of counsel and of power, the Spirit of knowledge and of the fear of the LORD-- 
3 and he will delight in the fear of the LORD. He will not judge by what he sees with his eyes, or decide what he hears with his ears; 
4 but with righteousness, he will judge the needy, with justice he will give decisions for the poor of the earth. 
Isaiah 11:1-4 (NIV)

A branch is a symbol of new life and new beginnings. Jesus was a descendant of King David and Christians assume that Jesus is this new branch.

The first Jesse trees were large carvings, tapestries or even stained glass windows put in Churches that helped illiterate people (people who can't read or write) to learn about the Bible from creation to the Christmas Story. But now Jesse Trees are used as a kind of Advent Calendar. You can use a normal Christmas tree or a banner in the shape of a tree. Each day through advent (or sometimes just on the four Sundays of Advent) a special decoration or ornament (similar to a Chrismon), that tells a story from the Bible, is hung on the tree.