Tree (graph theory)

A tree is a connected graph with no cycles. Between any two vertices in a tree, exactly one path exists.

Graph (discrete mathematics): A graph is a structure consisting of vertices and edges connecting pairs of vertices. A tree is a graph constrained by connectivity and the absence of cycles.
Spanning tree: A spanning tree is a tree containing every vertex of a connected graph. It preserves reachability while removing every cycle from the larger graph.
Tree traversal: Tree traversal is the systematic visiting of vertices in a tree. Traversal exploits unique paths to visit a tree without cycle-based ambiguity.
Unicyclic graph: A unicyclic graph is a connected graph containing exactly one cycle. It differs from a tree by the presence of one cycle.
Path (graph theory): A path is a sequence of distinct vertices joined by consecutive edges. Between every pair of vertices in a tree, exactly one path exists.
Leaf (graph theory): A leaf is a vertex of degree one in a graph. Every finite tree with at least two vertices has at least two leaves.
Breadth-first search: Breadth-first search explores a graph in order of increasing distance from a starting vertex. On a tree, it visits vertices by their distance from the chosen root.
Complete graph: A complete graph has an edge between every pair of distinct vertices. Complete graphs with at least three vertices contain cycles and many alternate paths.
Cycle (graph theory): A cycle is a closed path with no repeated vertices except its starting and ending vertex. A tree excludes cycles, making every route between vertices unique.
Degree (graph theory): The degree of a vertex is the number of edges incident to it. Vertex degrees in a finite tree sum to twice its number of edges.

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