A tree is a connected graph with no cycles. Between any two vertices, there is exactly one path.

Rooted Tree: is a graph G such that:

• G is connected
• G has no cycles
• G has exactly one vertex called the root of the tree Sibiling Nodes
• Nodes of the same parent nodes Leaf Nodes
• Nodes without children

Path from node $n_1$ to $n_k$

• Decending path
• Length of path is the number of edges Depth of vertex
• Length of unique descending path from root to v
• The root is at depth 0 Height of vertex v
• v is the length of the longest path from v to one of its descendant leaves

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Rooted Tree Recursively

• A graph with N nodes and N-1 edges
• Graph has
  • one root r
  • Zero or more non-empty subtrees, each of whose root is connected to r by an edge
• Use a stack for recursive tree algorithms

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Tree Traversal

• A traversal is the process for visiting all of the vertices of a tree
• Often uses recursion
• Each kind corresponds to an iterator type
• Iterators are implemented non-recursively
  • Preorder
  • Postorder
  • Levelorder

Preorder

1. Visit Vertex
2. Visit Child Vertices (Recursive)

==DFS should be used==

Traversal Example

void listAll (int depth = 0) const{
	printName(depth);
	if(isDir()){
	for each file c in this directory (for each child)
		c.listAll(depth + 1);
}

Postorder Traversal

1. Visit child vertices
2. Then visit vertex (Recursive)

==DFS should be used==

Traversal Example

int totalSize = sizeOfFile)(;)
if(isDir()){
	for each file c in this directory (for each child)
		totalSize += c.size();
	return totalSize;
}

Level Order Traversal

1. Visit all vertices in level
2. Starting with level one and increasing (Use a queue, not recursive)

==BFS should be used==

Inorder Traversal

1. Left Child
2. Vertex
3. Right Child (Recursive)

==DFS should be used==

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Binary Trees

A binary tree is a rooted tree in which no vertex has more than two children, left and right child nodes

struct Node {
	Obj element;
	Node* left;
	Node* right;
}

Complete Binary Tree: A binary tree is complete iff every layer except possibly the bottom is fully populated with vertices. In addition, all nodes at the bottom level must occupy the leftmost spots consecutively.

Satisfies

• $2^H ≤ n < 2^{H+1}$
• $2^2 ≤ 7 ≤ 2^{2+2}, 2^2 ≤ 4 < 2^{2+1}$

Why is this this important

This ==significantly reduces== the search algorithms time complexity. Rather than needing to search a descending path in ==N==, it will be in ==Log (N)==. Since the tree is balanced to a degree. We know the height. For instance if we had a billion nodes stored within a tree, if complete there would only be a depth of ~31, if it is not then the depth could lead to a stack overflow, due to recursive calls overloading the stack, possible 1 billion nodes called!

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Vector Representation

/* parent      */     v[k] = v[(k-1)/2]
/* left child  */     v[k] = v[2*k + 1]
/* right child */     v[k] = v[2*k + 2]

A vector is much easier to store, and we can use random access! It is a convenient way, but it necessitates the tree to be complete.