.. _Binary2:

.. raw:: html

   <script>ODSA.SETTINGS.DISP_MOD_COMP = true;ODSA.SETTINGS.MODULE_NAME = "Binary2";ODSA.SETTINGS.MODULE_LONG_NAME = "Binary Trees Part 2";ODSA.SETTINGS.MODULE_CHAPTER = "Slides"; ODSA.SETTINGS.BUILD_DATE = "2019-04-01 22:21:50"; ODSA.SETTINGS.BUILD_CMAP = true;JSAV_OPTIONS['lang']='en';JSAV_EXERCISE_OPTIONS['code']='java';</script>


.. |--| unicode:: U+2013   .. en dash
.. |---| unicode:: U+2014  .. em dash, trimming surrounding whitespace
   :trim:



.. odsalink:: AV/Binary/FullCompCON.css

.. odsalink:: AV/Binary/BSTCON.css
.. This file is part of the OpenDSA eTextbook project. See
.. http://algoviz.org/OpenDSA for more details.
.. Copyright (c) 2012-2013 by the OpenDSA Project Contributors, and
.. distributed under an MIT open source license.

.. avmetadata::
   :author: Cliff Shaffer


===================
Binary Trees Part 2
===================

Binary Trees Part 2
-------------------

Full and Complete Binary Trees
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   Full binary tree: Each node is either a leaf or internal node with
   exactly two non-empty children.

   Complete binary tree: If the height of the tree is :math:`d`,
   then all leaves except possibly level :math:`d` are completely
   full.
   The bottom level has all nodes to the left side.

   .. inlineav:: FullCompCON dgm
      :align: center


Full Binary Tree Theorem (1)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   **Theorem:** The number of leaves in a non-empty full binary tree
   is one more than the number of internal nodes.

   **Proof** (by Mathematical Induction):

   **Base case:** A full binary tree with 1 internal node must have
   two leaf nodes.

   **Induction Hypothesis:** Assume any full binary tree **T** containing
   :math:`n-1` internal nodes has :math:`n` leaves.


Full Binary Tree Theorem (2)
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
   **Induction Step:** Given tree **T** with :math:`n` internal nodes,
   pick internal node :math:`I` with two leaf children.
   Remove :math:`I`'s children, call resulting tree **T'**.

   By induction hypothesis, **T'** is a full binary tree with :math:`n`
   leaves.

   Restore :math:`I`'s two children.
   The number of internal nodes has now gone up by 1 to reach
   :math:`n`.
   The number of leaves has also gone up by 1.


Full Binary Tree Corollary
~~~~~~~~~~~~~~~~~~~~~~~~~~
   **Theorem:** The number of null pointers in a non-empty tree is one
   more than the number of nodes in the tree.

   **Proof:** Replace all null pointers with a pointer to an empty leaf
   node.  This is a full binary tree.


Dictionary
~~~~~~~~~~
   .. codeinclude:: Design/Dictionary
      :tag: DictionaryADT


.
~
   .


Dictionary (2)
~~~~~~~~~~~~~~
   * How can we implement a dictionary?

      * We know about array-based lists and linked lists.
      * They might be sorted or unsorted.
      * What are the pros and cons?


Binary Search Trees
~~~~~~~~~~~~~~~~~~~
   .. odsafig:: Images/BSTShape2.png
      :width: 500
      :align: center
      :capalign: justify
      :figwidth: 90%
      :alt: Two Binary Search Trees

BST as a Dictionary (1)
~~~~~~~~~~~~~~~~~~~~~~~
   .. codeinclude:: Binary/BST
      :tag: BSTa

BST as a Dictionary (2)
~~~~~~~~~~~~~~~~~~~~~~~
   .. codeinclude:: Binary/BST
      :tag: BSTb

BST ``findhelp``
~~~~~~~~~~~~~~~~
   .. inlineav:: BSTsearchCON ss
      :output: show


BST ``inserthelp``
~~~~~~~~~~~~~~~~~~
   .. inlineav:: BSTinsertCON ss
      :output: show


BST ``deletemax``
~~~~~~~~~~~~~~~~~
   .. inlineav:: BSTdeletemaxCON ss
      :output: show


BST ``removehelp``
~~~~~~~~~~~~~~~~~~
   .. inlineav:: BSTremoveCON ss
      :output: show


.
~
   .


BST Analysis
~~~~~~~~~~~~
   Find: :math:`O(d)`

   Insert: :math:`O(d)`

   Delete: :math:`O(d)`

   :math:`d =` depth of the tree

   :math:`d` is :math:`O(\log n)` if the tree is balanced.

   What is the worst case cost? When?

.. odsascript:: AV/Binary/FullCompCON.js
.. odsascript:: AV/Binary/BSTsearchCON.js
.. odsascript:: AV/Binary/BSTinsertCON.js
.. odsascript:: AV/Binary/BSTdeletemaxCON.js
.. odsascript:: AV/Binary/BSTremoveCON.js