A tree is a connected graph with no cycles. (a) Draw a bunch of trees. Conjecture a relationship between a the number of vertices and edges in any tree. (For instance, can you have a tree with 5 vertices and 7 edges?) (b) Explain why every tree with at least 3 vertices has at least one vertex with degree 1 (such a vertex is called a leaf. Hint: try a proof by contradiction. (c) Prove your conjecture from part (a) by induction on the number of vertices. Hint: For the inductive step, you will assume that your conjecture is true for all trees with k vertices, and show it is also true for an arbitrary tree with k + 1 vertices. So start with an arbitrary tree with k + 1 vertices. Consider what happens when you cut off a leaf and then let it regrow.
A tree is a connected graph with no cycles.
(a) Draw a bunch of trees. Conjecture a relationship between a the number of vertices and edges in any tree. (For instance, can you have a tree with 5 vertices and 7 edges?)
(b) Explain why every tree with at least 3 vertices has at least one vertex with degree 1 (such a vertex is called a leaf. Hint: try a proof by contradiction.
(c) Prove your conjecture from part (a) by induction on the number of vertices. Hint: For the inductive step, you will assume that your conjecture is true for all trees with k vertices, and show it is also true for an arbitrary tree with k + 1 vertices. So start with an arbitrary tree with k + 1 vertices. Consider what happens when you cut off a leaf and then let it regrow.
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