Tree Review

Objectives

We would like to :

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 2^h leaf nodes
Prove true from the base case:
     A binary tree of height 0 at most 2^h leaf nodes.

     Since a binary tree of height 0 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 2^k leaf nodes.

Prove true for k+1
     We know that a binary tree of height k has  at most 2^k leaf nodes.
     If each of these nodes is given two children, the number of leaf nodes will be 2*2^k = 2^k+1