Terminology

Learn all these terms:

• edge
• parent
• path
• ancestor
• descendent
• depth
• height
• level
• leaf
• internal
• full
• complete
• perfect

The Wonder and Joy of Binary Trees

• We will spend a lot of time thinking about binary trees.
• Why????
• Question for you:
  • What is the maximum number of nodes in a binary tree of height \(h\)?

  • \(N(h) \leq ??\)

    \(\displaystyle\sum_{i=1}^n c = cn\)	\(\displaystyle\sum_{i=1}^n i = \displaystyle\frac{n(n+1)}{2}\)
    \(\displaystyle\sum_{i=1}^n i^2 = \displaystyle\frac{n(n+1)(2n+1)}{6}\)	\(\displaystyle\sum_{i=0}^n r^i = \displaystyle\frac{1 - r^{n+1}}{1 - r}\)

1. \(\lfloor \log_2 h \rfloor\)
2. \(2h + 1\)
3. \(2^{h+1} - 1\)
4. \((2h^3 + 3n^2 + n) / 6\)
5. \(h!\)

The Wonder and Joy of Binary Trees

• We will spend a lot of time thinking about binary trees.
• Why????
• Question for you:
  • What is the maximum number of nodes in a binary tree of height \(h\)?
  • \(N(h) \leq 2^{h+1} - 1\)

Exercise #2

• What is the minimum height of a binary tree with \(n\) nodes?
• \(H(n) >= ??\)

1. \(\lceil \log_2 n \rceil\)
2. \(\lfloor \log_2 n \rfloor\)
3. \(\lfloor \frac{n}{2} \rfloor\)
4. \(n\)
5. \(2^n\)

Exercise #2

• What is the minimum height of a binary tree with \(n\) nodes?
• \(H(n) >= \lfloor \log_2 n \rfloor\)

Take Home Message

• In a linked list, we store a reference to one node, all other nodes are reachable by following \(O(n)\) references.

• In a binary tree, we store a reference to one node, all other nodes are reachable by following \(O(\log_2 n)\) references.

In-Order Traversal

public class Tree {

  public void printInOrder() {
    printInOrder(root);
  }

  private void printInOrder(Node node) {

    if (node != null) {
      printInOrder(node.getLeft());
      System.out.println(node.getItem());
      printInOrder(node.getRight());
    }

  }
}

In-Order Traversal (OO Version)

public class Node {

  public void printInOrder() {
    if (left != null) {
      left.printInOrder();
    }

    System.out.println(item);

    if (right != null) {
      right.printInOrder();
    }
  }

}

Pre-Order Traversal

  private void printPreOrder(Node node) {

    if (node != null) {
      System.out.println(node.getItem());
      printPreOrder(node.getLeft());
      printPreOrder(node.getRight());
    }

  }

Post-Order Traversal

   private void printPostOrder(Node node) {

    if (node != null) {
      printPostOrder(node.getLeft());
      printPostOrder(node.getRight());
      System.out.println(node.getItem());
    }

  }

Another Traversal

Trace the following traversal method…

  public void printMysteryOrder() {
    Queue<Node> queue = new LinkedList<>();
    
    queue.offer(root);  // enqueue
    
    while (!queue.isEmpty()) {
      Node curNode = queue.poll();  // dequeue
      System.out.println(curNode.getItem());
      
      if (curNode.getLeft() != null) {
        queue.offer(curNode.getLeft());
      }
      if (curNode.getRight() != null) {
        queue.offer(curNode.getRight());
      }
      
    }
    
  }