Tree Concepts

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Overview

A tree is a particular case of a graph.

Free Tree

A free tree is a connected, acyclic, undirected graph.

Graph FreeTree.png

For an undirected graph G = (V, E), the following statements are equivalent:

  1. G is a free tree.
  2. Any two vertices in G are connected by a unique simple path.
  3. G is connected, but if any edge is removed from E, the resulting graph is disconnected.
  4. G is connected and │E│ = │V│ - 1.
  5. G is acyclic and │E│ = │V│ - 1.
  6. G is acyclic, but if any edge is added to E, the resulting graph contains a cycle.

Forest

If the undirected graph is acyclic, but disconnected, it is a forest.

Graph Forest.png

Rooted Tree

A rooted tree is a free tree in which one of the vertices is distinguished from the others.

Tree Concepts.png

We call the distinguished vertex the root of the tree.

We often refer to the vertex of a rooted tree as a node of the tree.

For a node x in a rooted tree, we call any node y on the simple path from root to x an ancestor of x. If y is an ancestor of x, then x is a descendant of y. Every node is both an ancestor and descendant of itself. If y is an ancestor of x and x ≠ y, then we call y a proper ancestor of x, and x is a proper descendant of y.

The subtree rooted at x is the tree induced by descendants of x, rooted at x.

If the last edge on the simple path from the root of the tree to a node x is (y, x), then y is the parent of x and x is the child of y. The root is the only node in the tree with no parent.

If two nodes have the same parent, they are siblings.

A node with no children is a leaf or an external node. A non-leaf node is an internal node.


Search Tree

Binary Tree

Binary Search Tree

TODO Binary_Search_Tree_TODELETE