Three: Consider the following two definitions of a rooted undirected tree: Structural: A rooted tree of height n (n20) edges is a connected undirected graph without cycles where one node is designated as the root; and the longest path from the root to any node is of length n edges. (There could be more than one such path of length n.) Recursive A rooted tree of height 0 edges is a graph with one node (the root) and no edges A rooted tree of height n > 0 edges is a graph consisting of: The root node, One rooted sub-tree of height n-1, Zero or more additional rooted sub-trees of heightn-1, and For each sub-tree there is one edge between the root of the sub- tree and the root of the tree. All sub-trees are disjoint from each other. Prove that these two definitions are equivalent. (You may want to use induction for at least one direction of the proof.)