# Question: a tree is a connected acyclic graph in particular a...

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A tree is a connected acyclic graph. In particular, a single vertex is a tree. We define a Splitting Binary Tree, or SBTree for short, as either the lone vertex, or a tree with the following properties: 1. exactly one node of degree 2 (called the root). 2. every other node is of degree 3 or 1 (called internal nodes and leaves, respectively). For the case of one single vertex (see above), that vertex is considered to be a leaf.

(a) Show if an SBTree has more than one vertex, then the induced subgraph obtained by removing the unique root consists of two disconnected SBTrees. You may assume that by removing the root you obtain two separate connected componenents, so all you need to prove is that those two components are SBTrees.

(b) Prove that two SBTrees with the same number of leaves must also have the same total number of nodes. Hint: As a conjecture, guess an expression for the total number of nodes in terms of the number of leaves N(l). Then use induction to prove that it holds for all trees with the same l