You have a tree with horizontal branches and these occur at three levels. There is one branch at the top level, there are two branches at the 2nd level, there are 3 branches at the 1st level, and there is the ground below. A phoenix lives in your tree. At night, it roosts on the branch at the top level of your tree. In the morning, it descends to the ground. It can do this directly, or it can stop off anywhere along the way down. But it never moves horizontally, always down.
Use the following notation: T for the top branch; A and B for the branches on the second level; 1, 2, and 3 for the branches on the first level; and G for ground.
Draw a digraph representing the paths the phoenix can take from the top branch to the ground.
Determine how many unique paths the phoenix can take on its descent.
Imagine there are n levels to the tree with each level having one more branch than the one above it. How many paths will the phoenix have then?