# Graph Theory/Trees

A tree is a type of connected graph. An directed graph is a tree if it is connected, has no cycles and all vertices have at most one parent. An undirected graph is considered a tree if it is connected, has ${\displaystyle |V|-1}$ edges and is acyclic (a graph that satisfies any two of these properties satisfies all three).

 Exercise: Equivalent Definitions Show that the following are equivalent definitions for a tree: A graph with a minimal number of edges which is connected. A graph with maximal number of edges without a cycle. A graph with no cycle in which adding any edge creates a cycle. A graph with n nodes and n-1 edges that is connected. A graph in which any two nodes are connected by a unique path (path edges may only be traversed once). Hint: To keep the total proof short, put the definitions in a suitable order, and then prove A=>B=>C=>D=>E=>A. Take particular care over graphs with zero and one node. Additionally, A graph is connected and each edge is a bridge.
${\displaystyle tree}$

Acyclic and connected ${\displaystyle graph}$.
${\displaystyle forest}$ is acyclic graph. So, ${\displaystyle tree\subseteq forest}$

Meaning
• tree : minimum connected graph
• cycle : minimum 2-connected graph