A tree is a structure which grows from a single root node. By convention, the root node is usually placed at the top. Then, new nodes, referred as children nodes, can be attached to an existing node, referred as parent node, by edges. The root node has no parent and all other nodes have exactly one parent. A node may have any number of children nodes, including no child at all. No looping chain of nodes is permitted in this structure. For example, there are exactly $5$ different types of trees with $4$ nodes. They are shown below. The goal is to find the number of different types of tree with $n$ nodes where $n$ is a positive integer greater than $1$.