What are Trees?

Graph Theory · Axiom Academy

Discover the simplest way to connect things without creating loops. Click on each example to see how "tree" structures appear in everyday life. In graph theory, a tree is a connected graph with no cycles. Explore these examples: Try tracing from any vertex to any other vertex: Notice: There's always exactly one path between any two vertices. No shortcuts, no alternate routes! A tree must be connected (all vertices reachable) and have no cycles . Compare these graphs: Count the edges and vertices in the tree: What happens when you remove an edge from a tree? Click any edge below to remove it: A tree is a connected graph with no cycles - the simplest structure that keeps all vertices reachable from each other. A tree with n vertices always has exactly n - 1 edges. One more edge creates a cycle; one fewer edge disconnects the graph. Between any two vertices in a tree, there is exactly one path. This makes trees perfect for hierarchical structures and efficient searching. Every edge in a tree is critical - remove any single edge and the tree falls apart into disconnected pieces.

This is the written version of the interactive lesson above. See the full Graph Theory course.