Tree Review

Objectives

1. Review concepts associated with tree data structures.

Notes

• A tree is a directed graph
  • With one node known as the root.
  • Every node except the root has exactly one parent, or node pointing to it.
  • The root node has no parents.
  • Any node may have 0 or more children.
• Nodes with no children are known as leaf nodes.
• The height of a tree is the the longest path from the root to any node.
  • A tree with only a root has height 0.
  • A tree with a root with one child has height 1.
• The level of a node is the distance from the root to that node (or number of links.
• A complete tree is a tree where every level, except for the last, is full.
  • This makes sense when we limit the number of children a node can have.
• A binary tree is a tree where each node can have at most two children.
• A binary tree of height h will have at most 2^h leaf notes.

  By Induction:
      To Show: A binary tree of height h will have at most 2h leaf nodes
  Prove true from the base case:
       A binary tree of height 0 at most 2h leaf nodes.
  
       Since a binary tree of height 1 has only one node, the root, 
        which is also a leaf node, there are at most 1 = 20 leaf nodes.
  
  Assume true for k > 0
       A binary tree of height k has at most 2k leaf nodes.
  
  Prove true for k+1
       We know that a binary tree of height k has  at most 2k leaf nodes.
       If each of these nodes is given two children, the number of leaf nodes will be 2*2k = 2k+1