What Makes a Tree a Tree
Trees can come in many shapes and sizes. There are, however some constraints to making a valid tree.
 A tree has a single root
 A child has exactly one parent
 A tree is fully connected (IE a single tree)
 A tree has no cycles (IE no loops)
Valid Trees
Any combination of the following represents a valid tree:
Invalid Trees
Below are some examples of invalid trees.

A cycle Again, cycles are essentially loops that occur in our tree. In this example, we see that our leaf has two parents. One way to determine whether your data structure has a cycle is if there is more than one way to get from the root to any node.

A cycle Here we can see another cycle. In this case, the node immediately after the root has two parents, which is a clue that a cycle exists. Another test

Two Roots Trees must have a single root. In this instance, it may look like we have a tree with two roots. Working through this, we also see that the node in the center has two parents.

Two Trees This example would be considered two trees, not a tree with two parts. In this figure, we have two fully connected components. Since they are not connected to each other, this is not a single tree.