Questions Related to Subtree Order and their Elementary Solutions

We will consider two questions related to the order (number of vertices) of subtrees: 1) Consider the set of all subtrees as a partially ordered set, does the poset possess the Sperner property; 2) On the average order of subtrees compared to the order of the tree itself. The first question was raised by Penrice and second one by Jamison. Regardless of the pure graph theoretical nature of both topics, the solutions are elementary and share a common flavor. We aim to show through these two examples some of the basic ideas and approaches in graph theory, as well as to propose some open projects for potentially interested audience. Students are particularly welcome.