Trees2.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Binary Search Tree\n",
    "\n",
    "A binary search tree relies on the property that keys that are less than the parent are found in the left subtree, and keys that are greater than the parent are found in the right subtree."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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70uxs8668luvWU94zQMeGwAfN+Bx5TXVo3+eZ2QkxLg4mSbnXRV53GzBcI+9+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\ncmVzPTU5MDYsIHVuaXRzPTEWwuj5AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 1,
     "metadata": {
      "image/png": {
       "width": 300
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "from IPython.display import Image\n",
    "Image(filename='simpleBST.png', width=300) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "# https://www.laurentluce.com/posts/binary-search-tree-library-in-python/\n",
    "# The codes bleow are from the above git and web page."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "class Node(object):\n",
    "    \"\"\"Tree node: left and right child + data which can be any object\n",
    "\n",
    "    \"\"\"\n",
    "    def __init__(self, data):\n",
    "        \"\"\"Node constructor\n",
    "        \n",
    "        @param data node data object\n",
    "        \"\"\"\n",
    "        self.left = None\n",
    "        self.right = None\n",
    "        self.data = data\n",
    "\n",
    "    def insert(self, data):\n",
    "        \"\"\"Insert new node with data\n",
    "\n",
    "        @param data node data object to insert\n",
    "        \"\"\"\n",
    "        if self.data:\n",
    "            if data < self.data:\n",
    "                if self.left is None:\n",
    "                    self.left = Node(data)\n",
    "                else:\n",
    "                    self.left.insert(data) # recursive call\n",
    "            elif data > self.data:\n",
    "                if self.right is None:\n",
    "                    self.right = Node(data)\n",
    "                else:\n",
    "                    self.right.insert(data) # recursive call\n",
    "        else:\n",
    "            self.data = data\n",
    "\n",
    "    def lookup(self, data, parent = None):\n",
    "        \"\"\"Lookup node containing data\n",
    "\n",
    "        @param data node data object to look up\n",
    "        @param parent node's parent\n",
    "        @returns node and node's parent if found or None, None\n",
    "        \"\"\"\n",
    "        if data < self.data:\n",
    "            if self.left is None:\n",
    "                return None, None\n",
    "            return self.left.lookup(data, self)\n",
    "        elif data > self.data:\n",
    "            if self.right is None:\n",
    "                return None, None\n",
    "            return self.right.lookup(data, self)\n",
    "        else: # ==\n",
    "            return self, parent\n",
    "\n",
    "    def delete(self, data):\n",
    "        \"\"\"Delete node containing data\n",
    "\n",
    "        @param data node's content to delete\n",
    "        \"\"\"\n",
    "        # get node containing data\n",
    "        node, parent = self.lookup(data)\n",
    "        if node is not None:\n",
    "            children_count = node.children_count()\n",
    "            if children_count == 0:\n",
    "                # if node has no children, just remove it\n",
    "                if parent:\n",
    "                    if parent.left is node:\n",
    "                        parent.left = None\n",
    "                    else:\n",
    "                        parent.right = None\n",
    "                else: # why?\n",
    "                    self.data = None\n",
    "            elif children_count == 1:\n",
    "                # if node has 1 child\n",
    "                # replace node by its child\n",
    "                if node.left:\n",
    "                    n = node.left\n",
    "                else:\n",
    "                    n = node.right\n",
    "                if parent:\n",
    "                    if parent.left is node:\n",
    "                        parent.left = n\n",
    "                    else:\n",
    "                        parent.right = n\n",
    "                else:\n",
    "                    self.left = n.left\n",
    "                    self.right = n.right\n",
    "                    self.data = n.data\n",
    "            else:\n",
    "                # if node has 2 children\n",
    "                # find its successor\n",
    "                parent = node\n",
    "                successor = node.right\n",
    "                while successor.left:\n",
    "                    parent = successor\n",
    "                    successor = successor.left\n",
    "                # replace node data by its successor data\n",
    "                node.data = successor.data\n",
    "                # fix successor's parent node child\n",
    "                if parent.left == successor:\n",
    "                    parent.left = successor.right\n",
    "                else:\n",
    "                    parent.right = successor.right\n",
    "\n",
    "    def compare_trees(self, node):\n",
    "        \"\"\"Compare 2 trees\n",
    "\n",
    "        @param node tree to compare\n",
    "        @returns True if the tree passed is identical to this tree\n",
    "        \"\"\"\n",
    "        if node is None:\n",
    "            return False\n",
    "        if self.data != node.data:\n",
    "            return False\n",
    "        res = True\n",
    "        if self.left is None:\n",
    "            if node.left:\n",
    "                return False\n",
    "        else:\n",
    "            res = self.left.compare_trees(node.left)\n",
    "        if res is False:\n",
    "            return False\n",
    "        if self.right is None:\n",
    "            if node.right:\n",
    "                return False\n",
    "        else:\n",
    "            res = self.right.compare_trees(node.right)\n",
    "        return res\n",
    "\n",
    "    def print_tree(self):\n",
    "        \"\"\"Print tree content inorder\n",
    "\n",
    "        \"\"\"\n",
    "        if self.left:\n",
    "            self.left.print_tree()\n",
    "        print(self.data, end=\" \")\n",
    "        if self.right:\n",
    "            self.right.print_tree()\n",
    "\n",
    "    def children_count(self):\n",
    "        \"\"\"Return the number of children\n",
    "\n",
    "        @returns number of children: 0, 1, 2\n",
    "        \"\"\"\n",
    "        cnt = 0\n",
    "        if self.left:\n",
    "            cnt += 1\n",
    "        if self.right:\n",
    "            cnt += 1\n",
    "        return cnt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAAA3CAYAAABJnAVSAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAL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\nTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 4,
     "metadata": {
      "image/png": {
       "width": 200
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# __init__\n",
    "\n",
    "root = Node(8)\n",
    "Image(filename='bst1.png', width=200) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAMgAAACnCAYAAABU+hMRAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAL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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 5,
     "metadata": {
      "image/png": {
       "width": 200
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# insert()\n",
    "\n",
    "root.insert(3)\n",
    "root.insert(10)\n",
    "root.insert(1)\n",
    "Image(filename='bst2.png', width=200) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 6,
     "metadata": {
      "image/png": {
       "width": 300
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# insert()\n",
    "\n",
    "root.insert(6)\n",
    "root.insert(4)\n",
    "root.insert(7)\n",
    "root.insert(14)\n",
    "root.insert(13)\n",
    "Image(filename='bst3.png', width=300) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "# lookup()\n",
    "\n",
    "node, parent = root.lookup(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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Xrul2hf7+fktLS1tb28bGRub5MzAwYGdn19vb29raSvMhJgASgNSLCKaytWLF\nCowxtbu4biJJ8o9//GNTU5PmDVBMVKtWrWpraztx4gREOgDSXIBk5PNwSUkJQmj58uVjv5Stra2l\npSXDVl+XlJRgjMPDw+HVEoA0HyCZGew8PDw4HI5MJhv7pU6fPo0Q2rp1K5P8mThxIpfLbW9vhzAE\nQJoPkAwMdqdOndIvDZ6enmw2Wy+k0kHUvnPZ2dkQgwBIswKSgcFO78/51CZEr776KgPMqaiowBgH\nBQVBDAIgzQ1IpgW7vXv3IoSOHTum38uGhIQQBPHo0aOSkpJDhw6Zrj9OTk5sNtsoGyGZpwBI+gDJ\nkGCXk5Pz0Ucf9fT08Hg8Z2dnvV+/vr4eYywUCqlVx6ZlTk9Pz4cffjg4OLhlyxaEkEn/NgBIANLc\ng52vry9CiM1mG2LWZWdnJzUdVKH6+noTMmf37t1UDi6CIHx8fCASAZDmCSRDgp0i2QtCyN7evqSk\nRI8XP3r06JD5OkM2+qS5HBwcFC1ft24dRCIA0jyBZEKwKygoGPLdJyYm6reKhIQE5et7enqaijlV\nVVVDzAkNDYVgBECaIZBMCHYLFixQ9m7BggWGqCUqKkpRBcZY+7z4xhW1slohDofz4MEDiEcApBkC\nyYRgR3XTUjJcF4BcLnd1dVVOwWgS5lDdRopc/qbVuQNAApB6lMmvjb13715XVxf1t0gkKi8vN1BF\nBEHcvXuX2nADIUTlyKe5srKy+vr6qL8tLCyqq6tdXFxgmSoAaZ5A0j3rya+//nrgwIEbN240Nze3\nt7dbWFjY2dl5e3snJSWFhIQghN55552srCzqebi2ttbQ3jU0NHh5efX29pIk2dfX19/f//nnn3/9\n9dcSieS3335DCE2cONHBwWHZsmVvvfUWSZIGaoaW9U6dOvXnn39GCAkEgqqqKoh0AKRZA0nbx91D\nhw7Z29tTjeTxeBMmTPDw8HBwcBAIBNS+GxYWFnFxcdTIDkEQOqcJG61KS0upVrm7uxMEQaWjEIlE\nYrFYLBaLRCLqqyUIIiAgoLy8XL+1l5eX+/v7a1Nvc3Mz1U6hUPjo0SN4uwQgzRxIOga7q1evUlTZ\n2tomJSWpvtX39PQcPHjQz89PEbLHc/9gmUzm5eVFfYuLFy8uLi5WLVNUVBQdHU2SJMZ4zpw5elnG\nKJPJ5syZQyGlTb1WVlbU84VEIoFQBUACkLQLdocPHyYIgsfjpaWljVhYMcY/f/788Ul6U11dbW1t\njTFesmTJiGtcWltbo6OjMcY2NjY173cibAAABtdJREFUNTXjXC/lzPXr1yFaAZAAJO2C3YcffogQ\nEovFo1ort3btWoMOeylUVVXF4XDYbPa5c+e0Pys/P5/NZnM4nKqqKtOqFwRAMglIGgW7s2fPYoyD\ng4N1uCUePnwYGXg/wM7OTmtraz6fX1tbO9pza2pq+Hy+jY1NZ2enqdQLAiAZBiRdgl1nZyeHw3Fy\nctL54Z+aU/7FF18YqIXBwcFj6XUuLS0lCEKHyeLGqtfMBUAyD0i6BLulS5eSJNnS0jKWi0yZMsXa\n2toQzaM2ScnMzBzLRQ4cOIAQGtUySWPVCwIgmQckLYLdo0ePCIJ49913x3idO3fuYIx3796t9xZ6\neHjY29sP7+Bo5vRMnDhxVMsYh9R78uRJX19faqLDkJKfffbZ888/Tw32Y4xFItHGjRt1rtfMBUAy\nEkhaBLuVK1cSBKHNHuAVFRXUchN1BXx8fGxsbPTbvDt37ozqbhYbG+vm5jbsoYyMDISQlvu6q9bL\n5XIjIiLy8/NVHRAIBCtWrDh//nxnZ6dEIjl48KBAIJg7d64O9YIASEYCSYtgR81BH7GYXC63s7NL\nSkrSwBaVGFaHrlMNWrNmjfYLrfv6+thsdk5OzrBHZTIZxljLtDYa6tVmNnhpaSnGWId6QQAkI4E0\nfrCj5lWnpqaOWDImJsbDw0Ozs+3t7Qih5ORkPbbQwcHB1dVVy8I7d+60tLTUUEAsFjs5OY2xXm3Y\not6hdKjXzAVAMhVI4ycCKCsrQwjNnDlTc7Gvvvrq+++//+c//6m5mJWVFYfD+emnn/TYwra2Nu0X\n8X366afx8fEaCri4uEilUr3Xq6wnT55kZWWFh4eHhYXpUK+ZC4BkKpCk0dmqrKykZmBqNuutt946\ncuSIo6PjiBe0sLCQSCR6bGFvb++kSZO0KfnNN990dHTs27dPQxlnZ2dqy2Q91quc10zxt42NzcWL\nF3Wo18wFQDIVSOM/2VHRXbOJ4eHhoaGh1P6SI4rL5XZ3d+uxhQMDA4pEOpqVmpoaGRnJ4/E0lLG2\nth4YGNBvvUNG3Orr67dv397R0aGcRVL7es1cACRTgTT+k52npydC6O7du1SGHFV98MEHjY2N2ucF\n6+rq0m/qGJIkGxsbRyxWU1Pzyy+/fPfdd5qLNTU1aZlpR8t6h30x2bdvn5eX18aNG3Wo18wFQDIV\nSOM/2U2dOhUhpJqZXqGMjIzvvvuOw+FoecFnz545OzvrsYVcLrepqWnEYomJiVOmTKF+KhrU2Nio\n+U472nrVKS4uTi6X61CvmQuAZCyQRh/86uvrIwhi2bJlageMR9N4qndZvymqAwICRCKR5jIymYzF\nYhUUFIx4NSsrq4CAgDHWq80Xd/jwYT6fr0O9Zi4AkqlA0mKe3ZQpU2xtbUfRaPXOxsXFaTkdVHvt\n37+futVrKJOUlGRnZzfipajOby2ng2qoV9UBOzu71NTUysrKnp6empqa5ORkkiT37NmjQ70gAJKR\nQNIi2FHTqe/cuTN2tqytrX19ffXbPJlMRhDE66+/rqGMUCjUZlXQypUrWSyWlqkTVevV8DSRmZnp\n5OREDX4RBOHs7Jydna1bvSAAkpFA0iURwKiYUMcWtcDYEOmw4+LiWCzWqJKaqaqlpYXFYr355pv0\nrxcEQDIPSLoEuxMnTiCElGP/aCWRSLhcbkhIiCGaJ5PJOBxOcHDwWC4yc+ZMDofT3d1N/3pBACTz\ngKRR8s5p06bpnCdLLpc7Ojqy2eympiYDNY+6SyckJOh2OjXofvDgQVOpFwRAMgxIGgU7KgMql8sd\nducODWpvb588eTLGuLCw0KAtjI2NRQht2rRptCdSa8V1zltrrHrNXAAkw4Ck1x4UVVVVQqEQY7xv\n3z4tTykrK7OyssIYj8/IzuzZsxFCYWFhWj6Ed3d3h4aGIoTGmJrVWPWauQBIJgFJu93FpFKpm5sb\nQsjJyUnzjVEikYSHh2OMORzOhQsXxq2F7777LkKIw+Fs375d84vMtm3bqGRnOj/z06FeMxcAyRgg\nabpJ9p49e6jZ1SKR6JVXXjl69OjVq1elUmlFRcWpU6cSExPFYjHGGGO8cOHCMY4N6aDi4mJXV1eE\nEJvNDg4OTk9PP3fuXH19fX19/blz59LT02fOnEl9u66urlevXjX1ekEAJAOARLTFq6+vb/fu3d7e\n3tRm40M0ceLEVatW6Tcn4mh19uzZiIgIPp+v2jw+nx8RETGqjeboX6+ZC4A0dSDxoMblL3RQf39/\nRUVFRUXF/fv3HRwcpkyZEhwcTO0uThPdv3+/oqLi7t27CCE/P7+AgAB3d3cG12vmAiBNFEgTCHYg\nEAjEhKwnIBAINA76/4W9PhqvUr2AAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 8,
     "metadata": {
      "image/png": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# delete()\n",
    "\n",
    "root.delete(1)\n",
    "Image(filename='bst4.png', width=400) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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SxcUFrADCgXCahFFXV1eSJNva2sZ5HHt7exaLpWoJRpCy9u/fT3W6gRVAOBBu8mF0165d\nCKHMzMzxH6qsrAxjHBoaCgxpomnTpjEYjMbGRrACCAfCTTiMNjU1MRgMLy8vXR2QGjh24sQJYEhD\n81944QWwAggHwk04jOr8diGTyezs7LhcrkQiAYz0+aAEAsKBcAOE0QnqvLh69SpCaMGCBcCQJnJx\ncdFJtx0ICAfC9R1GJ/RV2ooVKxBChYWFwNCo0slLZBAQDoQbIIxO6MAuqVRqZWVlaWkJqYw0ETWk\n8fDhw2AFEA6Em0wY1cM0g6KiIoRQTEwMMKSJqBnKDQ0NYAUQDoSbQBjt7OxksVj29vYTXVFUVBRC\nqKSkBBgaVRUVFRjjOXPmgBVAOBBuAmF0zpw5+knB0tfXZ2FhYWNjY4a5GrUQlfI2Pz8frADCgXCj\nDqNffPGFPhMCfvnllwihNWvWAEOjikqFyeFwFPkcQUA4EG5EYXTfvn3Hjx+XSCQcDsfOzk6fN895\n8+ZhjEtLSxsaGiCtkXpdv34dYxwZGdna2hoQENDa2gqeAOFAuLGEUQaDgRCytrbGGF+7dk2fxonF\nYjabrchICAMk1SsuLo5Ki4sQWrduHRgChAPhYxUxETlMi4uLZTIZQqi7uxshVF1drc8Mqg8fPuRy\nub29vdS/H3zwAeQPVqUHDx78/PPPCKHBwUGE0OnTp8ETIBwIN4q0zSEhIUNqKSoq0tuNmiRJ5aot\nLS3hhqxKYWFhQ67UuXPnwBYgHAg3cKO+r69PsWoKpW3btunTuFOnTimWcKF04cIF4GlEyWQyHx8f\nZa9gRXsgHAg3fBj98MMPlU86LS1N/94VFhYqcwajI9Vz5uvrq/AKYwwpMIBwINzAYdTe3l5xxunp\n6Yby7ty5cwrOMMZ9fX3AkxrO/Pz8DBsXTEhAOBA+RGNbGfT27dvnz5+/d+/eo0ePrK2tPTw8goOD\nly9frihw586dWbNmUX9nZGTs3r3bgL3LRUVFy5Yto77gzp07P/roI8WmM2fOVFRUNDY2dnd3Ozo6\nenh4LFu2bObMmWbbEz84OBgYGPjLL79Q64N3dHSYp1dAOBCujVeaxNrGxsbFixdbWVkp9iJJUnEn\nJEly9uzZBQUFcrl88eLF1Id79uwxhrtQUVER1Y3F5/PlcnlBQYG/v7+ihx5jrNxbb2VltWTJkvv3\n75vtTTsgIICyory83Ky8AsKB8PF4NUoY7ezsXLJkCcYYY+zr65uZmdnU1KS89dSpUwsXLuRwOAgh\nd3d36gz27t1rPMZduHCBOitHR0eEEIfDWbhw4enTp5XXt2lqatq3b5+Pjw/1TZctW2a2E3v8/f2p\nwGEmXgHhQPj4vVIXRquqqiwtLTHGYWFh6kf/y2SyrVu3MplMhNCLL75obMa99dZbCCEmk7lt2zb1\ns02ampqoERJWVlbV1dVmCFlmZqb5eAWEA+E68UplGL1y5QqDwWCz2ZqnkxGLxV5eXvqcX6yJ1q9f\njxCaNm2aWCzWcJfi4mIWi8VgMK5cuWJWhJmVV0A4EK4rr5CqMMxkMgUCgXIDR0MtWrQIIZSTk2MM\nruXk5CCEoqOjx7pjU1OTnZ0di8XSwgETlVl5BYQD4Tr0auQwOmnSJA6H8/DhQ+1ONzAwkCCI8vJy\nw7pWWlqKMQ4ODtb6l8Zms/WQRNIYZG5eAeFAuA69GiGM7tq1C2N89epVrc+4v7/fzs5Oh2vMaidP\nT087O7vxLLpw5coVjPHu3btpD5lZeQWEA+G69QoN54PNZr/88svjPOlTp05RiQAM5VpBQQFC6OzZ\ns+M8TmRkJIfDkUqlNCbMrLwCwoFwnXs1NIympqZijNWk3jpx4oSvry81pE59lUKhcIJWSdREU6dO\nnTp1qqqtUqn0jTfeYLPZCCE2m71lyxZVJVtbWxFC27dvpzFkQ7zS8BJXVVVRb65NyysgHAhXc4k/\n/fTTadOmUcOhMMZ8Pn/z5s2jejX0KC4uLkKhUM0JsdnsyMjIwsLCUSFLTEzEGGv+RkyH6uzsxBir\nyWjr4+Pj7e1dUlLS09Nz4sQJJycnNUdzdHR0c3OjK2HDvdLkElN5xZOTk4eUMX6vgHAgXM0l5vF4\ncXFxFy5c6OnpaWtrO3DgAI/Hi4iIUO/VH47S19eHMd64caMmJzcqZPfv30cIGaTbJSMjAyGk6rVa\nbm7umLrVExISaDxnWY1Xai5xTEyMh4fH8DJG7hUQDoSP6RLL5fKysjKMsXqv/pDv68aNG3K5PCIi\nQieTWF1dXUmSpHKm6lllZWVMJtPZ2XnErf/4xz/+8pe/aH406l5069YtWs41Vu/ViPrqq68uXrz4\n448/mpxXQDgQPlZxudxRvfpDGK2rq0MI6TB/AZfLpXoT9KxHjx5Rs/dGVGtra2dnp42NDcaYIAiB\nQHDkyBE1R6Nyw/z222+0hEy9V8P15MmTt99++/Dhw9TUQ9PyCggHwseE+rFjx8LDw5VzP4/o1R/C\n6MOHDxFCL7zwgq6+gIWFRWdnp/6N6+rqUqxUM1wDAwO5ubnr1q1ra2t79OjRn/70p6SkpI8//lhV\neU9PT4TQgwcPaAmZeq+GKzw8PDQ0dOPGjaboFRAOhGsiaja9QCB49913McaXL19W79Ufwij1fNHY\n2KirL9DX18fn8/VvHJ/P7+vrU1MgJSXl0KFD9vb29vb2//znP//85z/v3LlTVWGqC8zJyYmWkI3q\nlbI++OCDlpYWRS4Mk/MKCAfCNRHV49nU1JSWltbd3U3NW1Pj1R/C6PTp0xFCt2/f1iFkDg4O+jdu\nypQpaowjCGLIKmC7du3q6upSVb6qqkphDv2k3qshysnJ+fbbb1kslol6BYQD4ZrL2dk5KyvryJEj\nN27cUO/VH8Lo3LlzqfWvddUlIZVKFQn+9KmgoCCpVNre3j7iVhsbmzEd7aeffsIYz507l5aQqfdq\niKRSaVhYGFaSogVkEl4B4UD4WBUfH0+tAqvOqyFv952cnJydnXUyHCQ1NRUhJBKJ9D/EgbJM1UJj\ncXFxqampyp+kpKTY2NhoN87Z1KXGK6TBcJAhZYzfKyAcCB8T4YcOHeJyueq9GnqUzZs3Y4w1IWPU\nMxh1nPOEysHBQdWI4s7OTmtr69TU1Pb29vb29i1bthAEkZubO2JhkUikfpwzDaTKq7GGUZPwCggH\nwtVcYoFAkJ6eXlNT09/fX19fn5KSQpKkYqUDVV4NPYpEImEymUuXLlXDlqoeWWVRbyHy8/MNZVxe\nXh5CqLi4eMStlZWVbm5uVFPUysoqOztb1XGio6NZLBa91wsb4pWGl3g4iCbhFRAOhKu5xLm5uUKh\nkPKNIAgnJ6e8vLxRvRqBj61bt2KMx5METCaT2dvbG3C6seLx28HBQX2Ca00mMNB7urEZegWEA+G6\n9QqNiAifz7e0tNS602f+/PkY42vXrhnWuGvXrmGMX3nlFe127+jo4PF4tra247HeVGRWXgHhQPhY\n9fjxYzVejdxYq62tZTAYTk5Oyis9aai1a9cihHbu3GkM3qWnpyOtlnwQiURCoZAkydraWrl5yKy8\nAsKBcB16pbLPq6CggCAIHo9XUVGhYWX9/f3U4I+YmBjj8S4mJoYa9KB5xtaysjIej0cQhAGzSRpE\n1OrBgYGB5uAVEG6GhG/YsGEiCFf3LvLq1ascDocgCE2WGM3KyqLm8KsahGFAUQNTuFzuqOvniMXi\nZcuWYYyZTKbBm2z6l2I03Ji84nA4JuoVEG5uhJ8/f97NzU3nhI8ypKO1tZX6aZEkGRYW9vnnnw/p\nGigrK1u9ejU1H27KlClFRUXGaV9GRgaVwtbGxiY+Pn7IA4hMJsvPzw8NDWUwGAghFovFZrPpnQ98\nuL744guE0Nq1a8+fPz9p0iQNvQoLC2tvbzfdbw2Em5t++OGH7777TreEI00qLisrCwkJUUwBZDAY\nFhYWin8xxh4eHocOHTJm7xwdHblcbm5urru7u2LKDYvFsrCwoMyi/g0JCamoqLh06RJCKCEhwXzY\nkkgkXC7Xzs5OEUQOHjyoiVf0+PpAuPlocHBQ54RjuYpRVCPq4sWLRUVFzc3N7e3tPB7Pzc3Nz89v\n06ZN2uWh0psOHz6cnJy8f//+v/71rwihZ8+eHT169Pbt2/fv35dIJPb29s7OzkuWLHnttdcUu/j7\n+9fU1LS1tVF3LdprwYIFJSUlV69ejYyMVP5cE6/oJCDc3KQTwscWRk1Rg4OD1tbWXC63o6ND871+\n++03Hx+fBQsWKOfIoqu++eab2NjYFStWnDlzBoGAcNBYRftneKr3/dSpU2PdMSoqCmNcU1NDb3/6\n+/stLS2trKzMs6cMCKc94XoQzZ9Gnz59amdn5+bmVl9fP9Z9Hz9+7ODg4OvrS6XGoquWL19+7ty5\noqIiarQTCAgHjVUEvb/e2rVrpVLpiRMntNh30qRJ69evv337dnFxMV39+f7778+dOxcVFQUxFAgH\naS06P402Nze7urrOnTtXOevqmDQwMGBlZWVnZ9fS0kLLPjWBQPD8+XORSGTkr1BAQLgxi7F79266\nfrfXXnuttbX1p59+sra21vJZnSAIgvjmm28cHByCgoJo5s+aNWvKy8u/+OKL2bNnwy8BCKcf4fA0\nOl6VlpbOnTt31apVBQUF4zyUQCCQSqVdXV0EQdDJn9DQ0Hnz5v3www/wMwDC6Uc4PI3qQJGRkRKJ\npLS0lMlkjvNQbm5uJ0+e7O3tffXVV2njT0BAwMDAQGVlJTX1BQSE04xwvYqW4w9OnjyJEHr//fd1\ndUBPT08mkymRSOjhD7U8snI+WhAQTifCYcCTDqTzRsqtW7dCQkJWrlx5+vRpUzfn9u3bs2fPDggI\nKC8vh8cIIJx+hOtfNOwK+eijj548eZKdna3Djp45c+aEhIQUFha2tLSUlpYeOnTIdP2Jjo4mSRLG\nuADhdCUcGvXaKz8/f/fu3f39/RwOx8nJSefHb2pqwhhbWloihFgslslNVfrwww/lcvm2bdsQQgcP\nHoSGGBBOJ8INK/qEUV9fX4QQ1d1+6dIl3R68p6cnNjZW+fbT1NRkQuZkZGQghKZMmUIQhI+PD3AP\nhNOMcAijupEiwxVCyN7e/ubNmzo8+CeffDLkKX7IOuBGLgcHB8WZb9iwAbgHwmlGOIRRHejcuXND\nIEhKStJtFYmJicrH9/T0NBVzamtrh5gTGhoK6APhtCEcwqhutGjRImUCFi1aNBG1REVFKarAGGu+\nnIthRa3VoxCLxbp//z6gD4TThnAIo7oR1S9OaeL6/mQymaurq6Kiw4cPm4Q5ysOzuVwu9HkB4TQj\n3OCiw4Cnu3fvPn36lPqbz+dXVlZO1Ogwgrhz5w61LA9CiFq8yMh17NgxqVRK/W1hYVFXV+fs7AwD\nVIBw2hBuDCKN/xT/+9//Zmdn37p1q729XSwWW1hYCAQCb2/v5OTkkJAQhFBWVpaiuVpdXa1YQmci\nZGlpWV1d7eXl9fz5c2r4+sDAwGefffb111+3tbU9fvwYITR58mQHB4c33njj7bffJsmJcljDev/+\n979Tf/B4vNraWoihQLipEA7jRnWjgwcP2tvbU+fJ4XAmTZrk4eHh4ODA4/GohagsLCzi4+Op19AE\nQZSVlennxMrKyqizcnd3p4ZAkyTJ5/NdXFxcXFz4fD7FFkEQ/v7+lZWVuq29srLSz89Pk3rb29sV\nv42HDx9C4wsINwnCoW9UN7p+/TqFl52dXXJy8vDuvP7+/gMHDsycOVNxPzh9+rTeTk8ikXh5eVEY\nLV26tKSkZHiZ4uLiJUuWkCSJMZ43b55OZitLJJJ58+ZRTGtSL5U/jcVitbW1QcwCwo2fcAijOtOh\nQ4cIguBwOPv27Ru1sGIgyCuvvDJkhfEJUl1dnY2NDcZ42bJlIpFIfWGRSLRkyRKMsa2tbX19vZ7r\npZwpLS2FsAWEGz/hEEZ1pg8//BAh5OLiMur1U9b69esn9A2mQrW1tSwWi8lknj9/XvO9CgsLmUwm\ni8Wqra01rXpBQDiQZmJh9OzZsxjj4OBgLe66VDKFFStWTNzp9fT02NjYcLnchoaGse5bX1/P5XJt\nbW17enpMpV4QEA6kmVgY7enpYbFYQqFQ65YLNQ3j888/n6AzDA4OHk83f1lZGUEQWswgMlS9ICAc\nSDOxMLpixQqSJDs6OsZzkOnTp9vYzSyDowAABitJREFU2EzE6VGrhuXm5o7nINnZ2QihMc2GNlS9\nICAcSDOxMPrw4UOCIN57771xHqe6uhpjnJGRofMz9PDwsLe3H9nEsQwmmzx58phmKw+p98SJE76+\nvtRomCElP/3002nTplEjUTDGfD5/8+bNWtcLAsL1QziEUZ1p9erVBEFIpdJRS1ZVVVGzG1UV8PHx\nsbW11e3pVVdXj+lGHRsb6+bmNuKmnJwchFBNTY129bLZ7MjIyMLCwuEO8Hi8uLi4Cxcu9PT0tLW1\nHThwgMfjRUREaFEvCAjXD+EQRnUpatrGqMVkMplAIEhOTlYD2d69exFCWvSRq9G6des0z9QglUqZ\nTGZ+fv6IWyUSCcZYw2x1aurVZOpEWVkZxliLekFAuH4IhzCqM1GTbdLT00ctGRMT4+HhoT6IiMVi\nhFBKSooOz9DBwcHV1VXDwjt37rSyslJTwMXFRSgUjrNeTcIo1QDUol4QEK4fwukho0hNUlFRgRAK\nCgpSX+yrr766ePHijz/+qL6YtbU1i8X69ddfdXiGXV1dms9G//jjjxMSEtQUcHZ27uzs1Hm9ynry\n5MmxY8fCw8PDwsK0qBcEhOuHcHrIKMJoTU0NNbRYfVx4++23Dx8+7OjoOOoBLSws2tradHiGz58/\nnzp1qiYlz5w5093drcglMaKcnJz6+/t1W69CGGOMsUAgePfddzHGly9f1qJeEBCuH8IhjOpM1I1L\n/VUMDw8PDQ2lFlgfVWw2u6+vT4dnODg4qMgepl7p6ekLFy7kcDhqytjY2AwODuq23iEvT5uamtLS\n0rq7u5XT/WpeLwgI1w/h9JBRJLny9PRECN25c4dKCzZcH3zwQUtLi+ZpFp8+farbjHAkSba0tIxa\nrL6+/vfff//222/VF2ttbdUwvZiG9Y7YqsrKyvLy8tq8ebMW9YKAcP0QDk+jOtOMGTMQQsOXDFIo\nJyfn22+/1TzN4rNnz5ycnHR4hmw2u7W1ddRiSUlJ06dPp34zatTS0qL+Zj7WelUpPj5eJpNpUS8I\nCNcP4RBGdaagoCCCINTc4qRSaVhYGFYS+v+dgMMLV1ZWymSyV199VYdn6OXldffuXfVlent7v//+\n+/379496tLt3744Koub1qtFnn33G5XK1qBcEhOuHcJrISEYMTJ8+3c7ObgwDtVSfeXx8vIbjnDUX\nhY767DXJyckCgWDUQ1FvGzQc56ym3uEOCASC9PT0mpqa/v7++vr6lJQUkiT37NmjRb0gIFw/hMO4\nUV2KmvlQXV09fshsbGx8fX11e3oSiYQgiDfffFNNGUtLS02m6K1evZrBYGiY43Z4vWruhbm5uUKh\nkHp+IQjCyckpLy9Pu3pBQLh+CIcwqmONCQ5VkFGZESZirYX4+HgGgzGmHJHD1dHRwWAw1qxZY/z1\ngoBwIM30wujx48cRQsoPUGNVW1sbm80OCQmZiNOTSCQsFis4OHg8BwkKCmKxWH19fcZfLwgIB9JM\nL4zK5fJZs2Zpne5QJpM5OjoymczW1tYJOj3qQSAxMVG73amxRwcOHDCVekFAOJBmemGUSr7NZrNH\nXEJLjcRi8QsvvIAxLioqmtAzjI2NRQht2bJlrDtSySa0Tl1uqHpBQDiQZmJhVC6X19bWWlpaYoyz\nsrI03KWiosLa2hpjrJ+Xg3PnzkUIhYWFadhy6evrCw0NRQiNMyu4oeoFAeFAmomFUblc3tnZ6ebm\nhhASCoXq771tbW3h4eEYYxaLdenSJb2d4XvvvYcQYrFYaWlp6lth27dvp3JHat1QMoZ6QUA4kGZi\nYZTSnj17qIkQfD5/+fLln3zyyfXr1zs7O6uqqk6ePJmUlOTi4kKNT46Ojh7n60UtVFJS4urqihBi\nMpnBwcGZmZnnz59vampqamo6f/58ZmZmUFAQhZerq+v169dNvV4QEA6kmV4YlcvlUqk0IyPD29ub\nIEaYbTV58uS33npLt8lrx6qzZ89GRkYqzxRSiMvlRkZGjmmVWuOvFwSEA2nDheWqF1oxHg0MDFRV\nVVVVVd27d8/BwWH69OnBwcHW1tbGc4b37t2rqqq6c+cOQmjmzJn+/v7u7u40rhcEhANpCplGGAWB\nQCBITQICgUD01P8BYkBjB0fSEyYAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 9,
     "metadata": {
      "image/png": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# delete()\n",
    "\n",
    "root.delete(14)\n",
    "Image(filename='bst5.png', width=400) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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o0L179548eSKVSl1dXb29vadPn/7222+rdgkLCystLa2vr6euWjbPW2+9lZ+f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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 10,
     "metadata": {
      "image/png": {
       "width": 400
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# delete()\n",
    "\n",
    "root.delete(3)\n",
    "Image(filename='bst6.png', width=400) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4 6 7 8 10 13 "
     ]
    }
   ],
   "source": [
    "# print_tree()\n",
    "\n",
    "root.print_tree()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "False"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "# compare_trees()\n",
    "\n",
    "root.compare_trees(root.left)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Binary Search Tree Analysis"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What is the **height** of a binary tree likely to be?\n",
    "\n",
    "The answer to this question depends on how the keys are added to the tree. If the keys are added in a random order, the height of the tree is going to be around $log_2 n$ where $n$ is the number of nodes in the tree.\n",
    "\n",
    "The number of nodes at any particular level is $2d$ where $d$ is the depth of the level. The total number of nodes in a perfectly **balanced binary tree** is $2^{h+1}−1$, where $h$ represents the height of the tree.\n",
    "\n",
    "In a balanced binary tree, the worst-case performance of *insert()* is $O(log_2 n)$, where $n$ is the number of nodes in the tree. \n",
    "\n",
    "Unfortunately it is possible to construct a search tree that has height $n$ simply by inserting the keys in sorted order! An example of such a tree is shown below. In this case the performance of the put method is $O(n)$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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M+pdOpysTDkfKXRQh/RKtD3eKTCYLXQ/CNAst7e3ts6qqqoZOmTK53GVdgL5l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\nD845bDYbZ4w5StmWAwUFICGnkwDkiTA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r9eLKK6+Ew+FEe/txvPDCCygUCpg3bx7Gjx9f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\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 13,
     "metadata": {
      "image/png": {
       "width": 300
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='skewedTree.png', width=300) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Q: how about delete() and "
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Balanced Tree\n",
    "\n",
    "As we learned, the performance of the binary search tree can degrade to $O(n)$ for operations like *insert()* when the tree becomes unbalanced. Now we look at a special kind of binary search tree that automatically makes sure that the tree remains balanced at all times. This tree is called an **AVL** tree and is named for its inventors: G.M. Adelson-Velskii and E.M. Landis.\n",
    "\n",
    "To implement our AVL tree we need to keep track of a **balance factor** for each node in the tree. \n",
    "\n",
    "$balanceFactor = height(leftSubTree) − height(rightSubTree)$\n",
    "\n",
    "We define a tree to be in balance if the balance factor is -1, 0, or 1. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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HZBh6TCmCCbHxBBmsMvSYUgRTUoMn+FQ1BbmeSBFMyoPE8qmBj5FSBJNSjuF8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\nRK5CYII=\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 15,
     "metadata": {
      "image/png": {
       "width": 300
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='unbalanced.png', width=300)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [],
   "source": [
    "# Again! Computing balance factor in a binary tree can be a 'divide-and-conquer' algorithm."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### The worst case of AVL tree"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AAdIIKvGuBtCR4/xAF63PueLBpCvDlKeVXJWtRcrzSnNlh+D4v5QOyhpDLUhR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\nMDYsIHVuaXRzPTEWwuj5AAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<IPython.core.display.Image object>"
      ]
     },
     "execution_count": 17,
     "metadata": {
      "image/png": {
       "width": 600
      }
     },
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Image(filename='worstAVL.png', width=600)\n",
    "\n",
    "# The following trees are still AVL tree, but they are the example of the worst cases of heights 0, 1, 2, and 3."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Looking at the total number of nodes in the above trees.\n",
    "\n",
    "Height 0: total number of the nodes is 1.\n",
    "\n",
    "Height 1: total number of the nodes is 1+1 = 2.\n",
    "\n",
    "Height 2: total number of the nodes is 1+1+2 = 4.\n",
    "\n",
    "Height 3: total number of the nodes is 1+4+2 = 7.\n",
    "\n",
    "Height 4: total number of the nodes is 1+7+4 = 12.\n",
    "\n",
    "The number of nodes in a tree of height $h$ ($N_h$) is:\n",
    "\n",
    "$ N_h = 1 + N_{h−1} + N_{h−2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "For your reference (i.e., you don't have to remember this):\n",
    "\n",
    "The height of an AVL tree with $N_h$ nodes satisfies\n",
    "\n",
    "$h < 1.4404 log_2 (N_h+2) - 0.328$\n",
    "\n",
    "or simply\n",
    "\n",
    "$ h < 2 log N_h$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Rebalance\n",
    "Please take a look at the AVL_rotation.pdf.\n",
    "\n",
    "An important property of AVL trees is that one rotation (single or double) is always sufficient after an insertion.\n",
    "\n",
    "However, deletion are more complicated. It is no longer true that the tree can be rebalanced with only one rotation after a deletion. There are cases where $O(log n)$ rotations are required, where $n$ is the number of nodes in the tree.\n",
    "\n",
    "Fortunately, each rotation requires only constant number of steps; thus, the worst-case running time of a deletion is still $O(log n)$."
   ]
  }
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