To tell if a graph is a tree, you check if it meets two fundamental requirements: it must be connected and contain no cycles.
The Core Definition of a Tree
A graph is formally defined as a tree if it satisfies specific structural properties. Based on standard definitions in graph theory, a graph is considered a tree if and only if:
- There is no cycle. A cycle is a path that starts and ends at the same vertex, visiting other vertices in between. Trees are acyclic.
- The graph is connected. This means that for any two vertices in the graph, there is a path connecting them. There are no isolated parts or disconnected components in a tree.
These two conditions are essential and sufficient to define a tree structure.
Condition 1: No Cycles
A key characteristic of a tree is the absence of cycles.
- What is a Cycle? In a graph, a cycle is a sequence of vertices and edges starting and ending at the same vertex, where no edge is repeated (and usually, no vertex is repeated except the start/end vertex). Think of it as a closed loop.
- Why Absence is Key: If a graph contains a cycle, you can find multiple paths between the same two vertices within that cycle. A tree structure, by contrast, guarantees a unique path between any two vertices. Removing any edge in a tree will disconnect it; this is not true in a graph with a cycle.
Example: A simple square shape formed by four vertices and four edges is not a tree because it contains a cycle. A straight line of vertices connected sequentially is a tree (specifically, a path graph).
Condition 2: Must Be Connected
The second crucial condition is that the graph must be connected.
- What is Connectivity? A graph is connected if there is a path between every pair of distinct vertices. Imagine starting at any vertex and being able to reach any other vertex by traversing edges.
- Why Connectivity is Key: Trees are structures where all components are linked together. A disconnected graph is essentially two or more separate graphs with no edges connecting vertices between them. A tree must be a single, cohesive unit.
Example: Two separate triangles are not a tree; they form two disconnected components. A single triangle is connected but not a tree because it has a cycle. A graph consisting of a few vertices linked together without forming loops is connected and could be a tree if it has no cycles.
Practical Ways to Check
To determine if a given graph is a tree, you can implement algorithms to check these two conditions:
- Checking for Connectivity:
- Perform a graph traversal (like Depth First Search - DFS or Breadth First Search - BFS) starting from an arbitrary vertex.
- After the traversal, check if all vertices in the graph were visited. If yes, the graph is connected.
- Checking for Cycles:
- During a traversal like DFS, keep track of the vertices visited and the path taken from the starting vertex.
- If you encounter a visited vertex that is not the immediate parent of the current vertex in the DFS tree, you have found a cycle.
- Alternatively, you can use algorithms specifically designed for cycle detection.
A common shortcut or alternative definition often used alongside the primary definition for graphs with a finite number of vertices ($n$) is:
- A graph with $n$ vertices is a tree if it is connected and has exactly $n-1$ edges.
- A graph with $n$ vertices is a tree if it is acyclic and has exactly $n-1$ edges.
If a graph satisfies any two of the properties (connected, acyclic, n-1 edges for n vertices), it is a tree (assuming it's a simple graph, meaning no loops or multiple edges between the same two vertices). However, the fundamental definition relies on the two conditions stated initially.
