Phase 1: Classify three graphs — which are valid trees?

Is This a Tree?

— Confession:I used to think I knew what a tree looked like. Then I saw a graph with n-1 edges and got it wrong. The problem is that “looks like a tree” is not a checkable definition. Two conditions make a tree — and each one, alone, is not enough.

Three graphs below. Before any explanation — classify each one: tree or not a tree? You have everything you need: node count and edge count are labeled. Trust your intuition, then see how it holds up.

A tree is a minimalist. It connects everyone at the party with the fewest possible handshakes. No redundant connections, no circular introductions. If anyone drops out, the party splits. Trees have no backup plan.

Graph A5 nodes, 4 edges
Graph B5 nodes, 5 edges
Graph C5 nodes, 3 edges