Tree

Definition

A tree is a connected graph with no cycles.

Connected means every node can be reached from every other node.

No cycles means there is no path that loops back to its starting point.

Tree = connected graph without loops.

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Simple Example

Example tree:

    A
   / \
  B   C
     / \
    D   E

The graph is connected.

There is exactly one path between any two nodes.

There are no cycles.

A tree branches without closing into a loop.

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Non-Example: Cycle

This is not a tree:

A -- B
|    |
D -- C

Why?

Because it contains a cycle:

A -- B -- C -- D -- A

A graph with a cycle is not a tree.

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Non-Example: Disconnected

This is not a tree:

A -- B

C -- D

Why?

Because the graph has two disconnected pieces.

There is no path from A to C.

A tree must be one connected structure.

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Tree As Hierarchy

Trees are often used to represent hierarchy.

Examples:

• folders inside folders
• categories and subcategories
• decision branches
• family-style ancestry structures
• expression structure

When a tree is given a starting node, it becomes a rooted tree.

Tree structure supports branching without circular dependency.

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Final Summary

• A tree is a connected graph with no cycles.
• Every node in a tree is reachable from every other node.
• A tree cannot contain loops.
• A disconnected graph is not a tree.
• Trees are useful for hierarchy and branching structure.