Paul: Überarbeitung im SLZ

Uebung08/Paul/Main.java

@@ -1,12 +1,22 @@
-import java.util.Random;
+import java.util.*;
 
 public class Main {
     public static void main(String[] args) {
-        RBTree<IntComparable> rbTree = new RBTree<>();
+
         Random random = new Random();
+        RBTree<IntComparable> rbTree = new RBTree<>();
+        ArrayList<Integer> numbers = new ArrayList<>();
+
+        // ein eingebauter Mechanismus, der sicherstellt, dass eine Zahl nicht 2-mal vorkommt, sonst würde der Baum nicht funktionieren
+        // Füllen der Liste mit Zahlen
+        for (int j = 1; j <= 100; j++) {
+            numbers.add(j);
+        }
+        //Collections.shuffle(numbers); // Liste mischen
+
+        for (int i = 0; i < 15; i++) {
+            int randomValue = numbers.get(i);
 
-        for (int i = 0; i <= 15; i++) {
-            int randomValue = random.nextInt(100);
             IntComparable key = new IntComparable(randomValue);
 
             // Füge das Element ein und aktualisiere die DOT-Datei nach jedem Schritt
@@ -19,4 +29,3 @@ public class Main {
         // ...
     }
 }   
-

Uebung08/Paul/RBTree.java

@@ -11,7 +11,7 @@ class RBTree<T extends Comparable<T>> {
 
     private class Node {
         T key;
-        Node left, right;
+        Node left, right, parent; // Elternknoten hinzugefügt
         boolean color;
 
         Node(T key, boolean color) {
@@ -22,38 +22,142 @@ class RBTree<T extends Comparable<T>> {
 
     private Node root;
 
+    private Node getUncle(Node parent) {
+        Node grandparent = parent.parent;
+
+        if (grandparent.left == parent) {
+            return grandparent.right; // right is the uncle
+
+        } else if (grandparent.right == parent) {
+            return grandparent.left; // left is the uncle
+
+        } else {
+            throw new IllegalStateException("Parent is not a child of its grandparent");
+        }
+    }
+
     public void insert(T key) {
-        root = insert(root, key);
+        root = insert(root, null, key); // Parent für die Wurzel ist null
         root.color = BLACK; // Ensure the root is always black
     }
 
-    private Node insert(Node root, T key) {
-        if (root == null) {
-            return new Node(key, RED); // New nodes are always red
-        }
+    private Node insert(Node root, Node parent, T key) {
+        Node node = root;
+        // Node parent = null;
 
-        int cmp = key.compareTo(root.key);
-        if (cmp < 0) {
-            root.left = insert(root.left, key);
-        } else if (cmp > 0) {
-            root.right = insert(root.right, key);
+        // Traverse the tree to the left or right depending on the key
+        while (node != null) {
+            parent = node;
+            if (key.compareTo(node.key) < 0) {
+                node = node.left;
+            } else if (key.compareTo(node.key) > 0) {
+                node = node.right;
+            } else {
+                throw new IllegalArgumentException("BST already contains a node with key " + key);
+            }
+        }
+        // Insert new node
+        Node newNode = new Node(key, RED);
+        if (parent == null) {
+            root = newNode;
+        } else if (key.compareTo(parent.key) < 0) {
+            parent.left = newNode;
         } else {
-            // Duplicate key, do not insert
-            return root;
+            parent.right = newNode;
+        }
+        newNode.parent = parent;
+        fixColorafterInsert(newNode);
+        return newNode;
+    }
+
+    /*
+     * Fix any Red-Black tree violations
+     * if (isRed(root.right) && !isRed(root.left)) {
+     * root = rotateLeft(root);
+     * }
+     * if (isRed(root.left) && isRed(root.left.left)) {
+     * root = rotateRight(root);
+     * }
+     * if (isRed(root.left) && isRed(root.right)) {
+     * flipColors(root);
+     * }
+     */
+
+    private void fixColorafterInsert(Node node) {
+        Node parent = node.parent;
+
+        // Case 1: no root, add node as new root, root is always black
+        if (parent == null) {
+            node.color = false;
+            return;
         }
 
-        // Fix any Red-Black tree violations
-        if (isRed(root.right) && !isRed(root.left)) {
-            root = rotateLeft(root);
+        // Parent is black --> nothing to do
+        if (parent.color == false) { // Abbildung S.312, aber beide Knoten sind schon schwarz gefärbt
+            return;
         }
-        if (isRed(root.left) && isRed(root.left.left)) {
-            root = rotateRight(root);
+
+        // From here on, parent(vater) is red
+        Node grandparent = parent.parent;
+
+        Node uncle = getUncle(parent); // Get the uncle (maybe nil, in which case its color is BLACK)
+
+        // Case 3 (parent uncle red, Abbildung S.313, 314): recolor parent, grandparent
+        // and uncle
+        if (uncle != null && uncle.color == true) {
+            parent.color = false;
+            uncle.color = false;
+            grandparent.color = true;
+
+            // Call recursively for grandparent, which is now red. fix recursively
+            fixColorafterInsert(grandparent);
         }
-        if (isRed(root.left) && isRed(root.right)) {
-            flipColors(root);
+
+        // Parent is left child of grandparent
+        // Case 4a: Uncle is black(nil is default black) and node is left --> right
+        // "inner child" of its grandparent
+        else if (parent == grandparent.left) {
+
+            Node newParent;
+
+            if (node == parent.right) {
+                newParent = rotateLeft(parent);
+                parent = newParent;
+
+                // Let "parent" point to the new root node of the rotated sub-tree.
+                // It will be recolored in the next step, which we're going to fall-through to.
+                // parent = node;
+            }
+
+            // Case 5a: Uncle is black and node is left->left "outer child" of its
+            // grandparent
+            rotateRight(grandparent);
+
+            // Recolor original parent and grandparent
+            parent.color = false;
+            grandparent.color = true;
         }
 
-        return root; // Return the updated root
+        // Parent is right child of grandparent
+        else if (parent == grandparent.right) {
+            // Case 4b: Uncle is black and node is right->left "inner child" of its
+            // grandparent
+            if (node == parent.left) {
+                rotateRight(parent);
+
+                // Let "parent" point to the new root node of the rotated sub-tree.
+                // It will be recolored in the next step, which we're going to fall-through to.
+                parent = node;
+            }
+
+            // Case 5b: Uncle is black and node is right->right "outer child" of its
+            // grandparent
+            rotateLeft(grandparent);
+
+            // Recolor original parent and grandparent
+            parent.color = false;
+            grandparent.color = true;
+        }
     }
 
     private boolean isRed(Node node) {
@@ -63,39 +167,107 @@ class RBTree<T extends Comparable<T>> {
         return node.color == RED;
     }
 
-    private Node rotateLeft(Node h) {
-        Node x = h.right;
-        h.right = x.left;
-        x.left = h;
-        x.color = h.color;
-        h.color = RED;
-        return x;
+    private Node rotateLeft(Node node) {
+        Node rightChild = node.right;
+        Node parent = node.parent;
+
+        node.right = rightChild.left;
+        if (rightChild.left != null) {
+            rightChild.left.parent = node;
+        }
+
+        rightChild.left = node;
+        node.parent = rightChild;
+
+        rightChild.parent = parent;
+        if (parent == null) {
+            root = rightChild;
+        } else if (node == parent.left) {
+            parent.left = rightChild;
+        } else if (node == parent.right) {
+            parent.right = rightChild;
+        } else {
+            throw new IllegalStateException("Node is not a child of its parent");
+        }
+
+        if (rightChild != null) {
+            rightChild.parent = parent;
+        }
+
+        RootBlack();
+
+        return rightChild;
     }
 
-    private Node rotateRight(Node h) {
-        Node x = h.left;
-        h.left = x.right;
-        x.right = h;
-        x.color = h.color;
-        h.color = RED;
-        return x;
+    private Node rotateRight(Node node) {
+        Node leftChild = node.left;
+        Node parent = node.parent;
+
+        node.left = leftChild.right;
+        if (leftChild.right != null) {
+            leftChild.right.parent = node;
+        }
+
+        leftChild.right = node;
+        node.parent = leftChild;
+
+        leftChild.parent = parent;
+        if (parent == null) {
+            root = leftChild;
+        } else if (node == parent.left) {
+            parent.left = leftChild;
+        } else if (node == parent.right) {
+            parent.right = leftChild;
+        } else {
+            throw new IllegalStateException("Node is not a child of its parent");
+        }
+
+        if (leftChild != null) {
+            leftChild.parent = parent;
+        }
+
+        RootBlack();
+
+        return leftChild;
     }
 
-    private void flipColors(Node h) {
-        h.color = RED;
-        h.left.color = BLACK;
-        h.right.color = BLACK;
+    private void flipColors(Node node) {
+        node.color = !node.color;
+        node.left.color = !node.left.color;
+        node.right.color = !node.right.color;
     }
 
-    public void printDOTAfterInsert(String filename, T key) {
+    private void RootBlack() {
+        root.color = BLACK;
+    }
+
+    /*
+     * public void printDOTAfterInsert(String filename, T key) {
+     * try (FileWriter writer = new FileWriter(filename)) {
+     * writer.write("digraph G {\n");
+     * writer.write("\tnode [style=filled, color=black, shape=circle, width=.6,\n");
+     * writer.write("\t\tfontname=Helvetica, fontweight=bold, fontcolor=white,\n");
+     * writer.write("\t\tfontsize=24, fixedsize=true];\n");
+     * 
+     * root = insert(root, null, key); // Füge das Element ein und aktualisiere den
+     * Baum
+     * printDOTRecursive(root, writer);
+     * 
+     * writer.write("}\n");
+     * } catch (IOException e) {
+     * e.printStackTrace();
+     * }
+     * }
+     */
+
+     public void printDOTAfterInsert(String filename, T key) {
         try (FileWriter writer = new FileWriter(filename)) {
             writer.write("digraph G {\n");
-            writer.write("\tgraph [ratio=.48];\n");
-            writer.write("\tnode [style=filled, color=black, shape=circle, width=.6 \n");
-            writer.write("\t\tfontname=Helvetica, fontweight=bold, fontcolor=white, \n");
+            writer.write("\tnode [style=filled, color=black, shape=circle, width=.6,\n");
+            writer.write("\t\tfontname=Helvetica, fontweight=bold, fontcolor=white,\n");
             writer.write("\t\tfontsize=24, fixedsize=true];\n");
     
-            root = insert(root, key); // Füge das Element ein und aktualisiere den Baum
+            // Hier wird der gesamte Baum gezeichnet, nicht nur der neu eingefügte Schlüssel
             printDOTRecursive(root, writer);
     
             writer.write("}\n");
@@ -106,22 +278,24 @@ class RBTree<T extends Comparable<T>> {
     
     private void printDOTRecursive(Node node, FileWriter writer) throws IOException {
         if (node != null) {
-            String keyString = node.key.toString().replaceAll("[^a-zA-Z0-9]", "_"); // Ersetze ungültige Zeichen
-    
-            // Setze Farben basierend auf der Rot-Schwarz-Eigenschaft
+            String keyString = node.key.toString().replaceAll("[^a-zA-Z0-9]", "_");
             String fillColor = (node.color == RED) ? "red" : "black";
-            String fontColor = (node.color == RED) ? "white" : "white"; // Ändern Sie dies, wenn Sie schwarzen Text auf rotem Hintergrund möchten
+            String fontColor = (node.color == RED) ? "white" : "white";
     
-            // Print current node mit Farbinformationen
             writer.write("\t\"" + keyString + "\" [fillcolor=" + fillColor + ", fontcolor=" + fontColor + "];\n");
     
+            // Print parent link
+            if (node.parent != null) {
+                String parentKeyString = node.parent.key.toString().replaceAll("[^a-zA-Z0-9]", "_");
+                writer.write("\t\"" + parentKeyString + "\" -> \"" + keyString + "\" [style=dotted];\n");
+            }
+    
             // Print left child
             if (node.left != null) {
                 String leftKeyString = node.left.key.toString().replaceAll("[^a-zA-Z0-9]", "_");
                 writer.write("\t\"" + keyString + "\" -> \"" + leftKeyString + "\";\n");
                 printDOTRecursive(node.left, writer);
             } else {
-                // Füge ein NIL-Blatt für leere linke Zweige hinzu
                 String leftNilKeyString = keyString + "_NIL_L";
                 writer.write("\t\"" + leftNilKeyString + "\" [shape=plaintext, label=\"NIL\", fontsize=16];\n");
                 writer.write("\t\"" + keyString + "\" -> \"" + leftNilKeyString + "\";\n");
@@ -133,12 +307,69 @@ class RBTree<T extends Comparable<T>> {
                 writer.write("\t\"" + keyString + "\" -> \"" + rightKeyString + "\";\n");
                 printDOTRecursive(node.right, writer);
             } else {
-                // Füge ein NIL-Blatt für leere rechte Zweige hinzu
                 String rightNilKeyString = keyString + "_NIL_R";
                 writer.write("\t\"" + rightNilKeyString + "\" [shape=plaintext, label=\"NIL\", fontsize=16];\n");
                 writer.write("\t\"" + keyString + "\" -> \"" + rightNilKeyString + "\";\n");
             }
         }
     }
-}
     
+
+    /*public void printDOTAfterInsert(String filename, T key) {
+        try (FileWriter writer = new FileWriter(filename)) {
+            writer.write("digraph G {\n");
+            writer.write("\tnode [style=filled, color=black, shape=circle, width=.6,\n");
+            writer.write("\t\tfontname=Helvetica, fontweight=bold, fontcolor=white,\n");
+            writer.write("\t\tfontsize=24, fixedsize=true];\n");
+
+            // Hier wird der gesamte Baum gezeichnet, nicht nur der neu eingefügte Schlüssel
+            printDOTRecursive(root, writer);
+
+            writer.write("}\n");
+        } catch (IOException e) {
+            e.printStackTrace();
+        }
+    }
+
+    private void printDOTRecursive(Node node, FileWriter writer) throws IOException {
+        if (node != null) {
+            String keyString = node.key.toString().replaceAll("[^a-zA-Z0-9]", "_");
+            String fillColor = (node.color == RED) ? "red" : "black";
+            String fontColor = (node.color == RED) ? "white" : "white";
+
+            writer.write("\t\"" + keyString + "\" [fillcolor=" + fillColor + ", fontcolor=" + fontColor + "];\n");
+
+            /*
+             * Print parent link
+             * if (node.parent != null) {
+             * String parentKeyString =
+             * node.parent.key.toString().replaceAll("[^a-zA-Z0-9]", "_");
+             * writer.write("\t\"" + parentKeyString + "\" -> \"" + keyString +
+             * "\" [style=dotted];\n");
+             * }
+             
+
+            // Print left child
+            if (node.left != null) {
+                String leftKeyString = node.left.key.toString().replaceAll("[^a-zA-Z0-9]", "_");
+                writer.write("\t\"" + keyString + "\" -> \"" + leftKeyString + "\";\n");
+                printDOTRecursive(node.left, writer);
+            } else {
+                String leftNilKeyString = keyString + "_NIL_L";
+                writer.write("\t\"" + leftNilKeyString + "\" [shape=plaintext, label=\"NIL\", fontsize=16];\n");
+                writer.write("\t\"" + keyString + "\" -> \"" + leftNilKeyString + "\";\n");
+            }
+
+            // Print right child
+            if (node.right != null) {
+                String rightKeyString = node.right.key.toString().replaceAll("[^a-zA-Z0-9]", "_");
+                writer.write("\t\"" + keyString + "\" -> \"" + rightKeyString + "\";\n");
+                printDOTRecursive(node.right, writer);
+            } else {
+                String rightNilKeyString = keyString + "_NIL_R";
+                writer.write("\t\"" + rightNilKeyString + "\" [shape=plaintext, label=\"NIL\", fontsize=16];\n");
+                writer.write("\t\"" + keyString + "\" -> \"" + rightNilKeyString + "\";\n");
+            }
+        }
+    */
+}

output_step_0.dot

@@ -1,11 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"56" -> "56_NIL_L";
-	"56_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"56" -> "56_NIL_R";
 }

output_step_1.dot

@@ -1,15 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "33";
-	"33" [fillcolor=red, fontcolor=white];
-	"33_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_L";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"56" -> "56_NIL_R";
 }

output_step_10.dot

@@ -1,51 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=black, fontcolor=white];
-	"25" -> "15";
-	"15" [fillcolor=red, fontcolor=white];
-	"15" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_L";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"15" -> "19";
-	"19" [fillcolor=black, fontcolor=white];
-	"19_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_L";
-	"19_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"69" -> "98";
-	"98" [fillcolor=black, fontcolor=white];
-	"98" -> "89";
-	"89" [fillcolor=red, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
-	"98_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"98" -> "98_NIL_R";
 }

output_step_11.dot

@@ -1,55 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=black, fontcolor=white];
-	"25" -> "15";
-	"15" [fillcolor=red, fontcolor=white];
-	"15" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11" -> "8";
-	"8" [fillcolor=red, fontcolor=white];
-	"8_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_L";
-	"8_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_R";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"15" -> "19";
-	"19" [fillcolor=black, fontcolor=white];
-	"19_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_L";
-	"19_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"69" -> "98";
-	"98" [fillcolor=black, fontcolor=white];
-	"98" -> "89";
-	"89" [fillcolor=red, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
-	"98_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"98" -> "98_NIL_R";
 }

output_step_12.dot

@@ -1,55 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=black, fontcolor=white];
-	"25" -> "15";
-	"15" [fillcolor=red, fontcolor=white];
-	"15" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11" -> "8";
-	"8" [fillcolor=red, fontcolor=white];
-	"8_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_L";
-	"8_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_R";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"15" -> "19";
-	"19" [fillcolor=black, fontcolor=white];
-	"19_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_L";
-	"19_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"69" -> "98";
-	"98" [fillcolor=black, fontcolor=white];
-	"98" -> "89";
-	"89" [fillcolor=red, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
-	"98_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"98" -> "98_NIL_R";
 }

output_step_13.dot

@@ -1,59 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=black, fontcolor=white];
-	"25" -> "15";
-	"15" [fillcolor=red, fontcolor=white];
-	"15" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11" -> "8";
-	"8" [fillcolor=red, fontcolor=white];
-	"8_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_L";
-	"8_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_R";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"15" -> "19";
-	"19" [fillcolor=black, fontcolor=white];
-	"19_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_L";
-	"19_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "64";
-	"64" [fillcolor=black, fontcolor=white];
-	"64" -> "63";
-	"63" [fillcolor=red, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"64_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"64" -> "64_NIL_R";
-	"69" -> "98";
-	"98" [fillcolor=black, fontcolor=white];
-	"98" -> "89";
-	"89" [fillcolor=red, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
-	"98_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"98" -> "98_NIL_R";
 }

output_step_14.dot

@@ -1,63 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=black, fontcolor=white];
-	"25" -> "15";
-	"15" [fillcolor=red, fontcolor=white];
-	"15" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11" -> "8";
-	"8" [fillcolor=red, fontcolor=white];
-	"8_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_L";
-	"8_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_R";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"15" -> "19";
-	"19" [fillcolor=black, fontcolor=white];
-	"19" -> "18";
-	"18" [fillcolor=red, fontcolor=white];
-	"18_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"18" -> "18_NIL_L";
-	"18_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"18" -> "18_NIL_R";
-	"19_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "64";
-	"64" [fillcolor=black, fontcolor=white];
-	"64" -> "63";
-	"63" [fillcolor=red, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"64_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"64" -> "64_NIL_R";
-	"69" -> "98";
-	"98" [fillcolor=black, fontcolor=white];
-	"98" -> "89";
-	"89" [fillcolor=red, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
-	"98_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"98" -> "98_NIL_R";
 }

output_step_15.dot

@@ -1,67 +1,58 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=red, fontcolor=white];
-	"25" -> "15";
-	"15" [fillcolor=black, fontcolor=white];
-	"15" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11" -> "8";
-	"8" [fillcolor=red, fontcolor=white];
-	"8_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_L";
-	"8_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"8" -> "8_NIL_R";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"15" -> "19";
-	"19" [fillcolor=black, fontcolor=white];
-	"19" -> "18";
-	"18" [fillcolor=red, fontcolor=white];
-	"18_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"18" -> "18_NIL_L";
-	"18_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"18" -> "18_NIL_R";
-	"19_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"19" -> "19_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=black, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33" -> "49";
-	"49" [fillcolor=black, fontcolor=white];
-	"49_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"49" -> "49_NIL_L";
-	"49_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"49" -> "49_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "64";
-	"64" [fillcolor=black, fontcolor=white];
-	"64" -> "63";
-	"63" [fillcolor=red, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"64_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"64" -> "64_NIL_R";
-	"69" -> "98";
+	"75" [fillcolor=black, fontcolor=white];
+	"75" -> "73";
+	"73" [fillcolor=red, fontcolor=white];
+	"73" -> "51";
+	"51" [fillcolor=black, fontcolor=white];
+	"51" -> "39";
+	"39" [fillcolor=black, fontcolor=white];
+	"39" -> "38";
+	"38" [fillcolor=red, fontcolor=white];
+	"38" -> "34";
+	"34" [fillcolor=black, fontcolor=white];
+	"34" -> "28";
+	"28" [fillcolor=black, fontcolor=white];
+	"28" -> "13";
+	"13" [fillcolor=red, fontcolor=white];
+	"13_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
+	"13" -> "13_NIL_L";
+	"13_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
+	"13" -> "13_NIL_R";
+	"28_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
+	"28" -> "28_NIL_R";
+	"34" -> "37";
+	"37" [fillcolor=black, fontcolor=white];
+	"37_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
+	"37" -> "37_NIL_L";
+	"37_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
+	"37" -> "37_NIL_R";
+	"38_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
+	"38" -> "38_NIL_R";
+	"39_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
+	"39" -> "39_NIL_R";
+	"51_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
+	"51" -> "51_NIL_R";
+	"73_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
+	"73" -> "73_NIL_R";
+	"75" -> "77";
+	"77" [fillcolor=black, fontcolor=white];
+	"77_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
+	"77" -> "77_NIL_L";
+	"77" -> "80";
+	"80" [fillcolor=black, fontcolor=white];
+	"80_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
+	"80" -> "80_NIL_L";
+	"80" -> "98";
 	"98" [fillcolor=black, fontcolor=white];
-	"98" -> "89";
-	"89" [fillcolor=red, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
+	"98" -> "81";
+	"81" [fillcolor=black, fontcolor=white];
+	"81_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
+	"81" -> "81_NIL_L";
+	"81_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
+	"81" -> "81_NIL_R";
 	"98_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
 	"98" -> "98_NIL_R";
 }

output_step_2.dot

@@ -1,19 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_L";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
 }

output_step_3.dot

@@ -1,23 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "25";
-	"25" [fillcolor=red, fontcolor=white];
-	"25_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"25" -> "25_NIL_L";
-	"25_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"25" -> "25_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
 }

output_step_4.dot

@@ -1,27 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=red, fontcolor=white];
-	"25" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_L";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_L";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
 }

output_step_5.dot

@@ -1,31 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=red, fontcolor=white];
-	"25" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_L";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_L";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "89";
-	"89" [fillcolor=black, fontcolor=white];
-	"89" -> "63";
-	"63" [fillcolor=red, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
 }

output_step_6.dot

@@ -1,35 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=red, fontcolor=white];
-	"25" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_L";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "89";
-	"89" [fillcolor=black, fontcolor=white];
-	"89" -> "63";
-	"63" [fillcolor=red, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
 }

output_step_7.dot

@@ -1,39 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=black, fontcolor=white];
-	"25" -> "11";
-	"11" [fillcolor=black, fontcolor=white];
-	"11_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_L";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"69" -> "89";
-	"89" [fillcolor=black, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
 }

output_step_8.dot

@@ -1,43 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=black, fontcolor=white];
-	"25" -> "15";
-	"15" [fillcolor=black, fontcolor=white];
-	"15" -> "11";
-	"11" [fillcolor=red, fontcolor=white];
-	"11_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_L";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"15_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"15" -> "15_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"69" -> "89";
-	"89" [fillcolor=black, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
 }

output_step_9.dot

@@ -1,47 +1,5 @@
 digraph G {
-	graph [ratio=.48];
-	node [style=filled, color=black, shape=circle, width=.6 
+	node [style=filled, color=black, shape=circle, width=.6,
 		fontname=Helvetica, fontweight=bold, fontcolor=white,
 		fontsize=24, fixedsize=true];
-	"56" [fillcolor=red, fontcolor=white];
-	"56" -> "25";
-	"25" [fillcolor=black, fontcolor=white];
-	"25" -> "15";
-	"15" [fillcolor=black, fontcolor=white];
-	"15" -> "11";
-	"11" [fillcolor=red, fontcolor=white];
-	"11_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_L";
-	"11_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"11" -> "11_NIL_R";
-	"15_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"15" -> "15_NIL_R";
-	"25" -> "33";
-	"33" [fillcolor=black, fontcolor=white];
-	"33" -> "27";
-	"27" [fillcolor=red, fontcolor=white];
-	"27_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_L";
-	"27_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"27" -> "27_NIL_R";
-	"33_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"33" -> "33_NIL_R";
-	"56" -> "69";
-	"69" [fillcolor=black, fontcolor=white];
-	"69" -> "63";
-	"63" [fillcolor=black, fontcolor=white];
-	"63_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_L";
-	"63_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"63" -> "63_NIL_R";
-	"69" -> "98";
-	"98" [fillcolor=black, fontcolor=white];
-	"98" -> "89";
-	"89" [fillcolor=red, fontcolor=white];
-	"89_NIL_L" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_L";
-	"89_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"89" -> "89_NIL_R";
-	"98_NIL_R" [shape=plaintext, label="NIL", fontsize=16];
-	"98" -> "98_NIL_R";
 }