School of Informatics, University of Edinburgh Computer Science 1 Ah [03 04]


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1 School of Informatics, University of Edinburgh Computer Science 1 Ah [03 04] CS1Ah Lecture Note 25 Binary Search Trees This lecture studies binary trees in more detail, in particular their application as search trees. Binary search trees store data in a way that makes contains, add, and remove operations efficient. They can be used to represent finite sets or partial maps The binary search tree property A binary search tree stores data elements in its nodes, and exploits an ordering on those data elements. The ordering is used to enforce the binary search tree property: for each node n, the labels in n s left subtree T L are all less than the label of n, and the labels in n s right subtree T R are all greater than the label of n. In particular, this means that T L and T R also satisfy the property. Here s an example of a tree which satisfies the property, using integers as labels: An inorder traversal of the tree will visit the nodes in ascending order. Notice that the search tree property doesn t restrict the shape of the tree. The contents of the tree above could be stored in either of these essentially equivalent search trees:
2 25.2 Implementing search trees Now we will consider implementing search trees. The application will be a data type which implements a collection, similar to the application of arrays we illustrated in CS1Ah Lecture Note 17, but using a tree to hold the collection instead of an array. To describe the methods we will implement, we will use a Java interface. An interface in Java describes the methods which should be implemented by a class. For each method, the header is given: as usual, the header shows the result type, method name, and argument names and types. But no method body describing the method s behaviour is provided. An interface serves as a basic specification of some behaviour; to be useful it must be supplemented by careful documentation. The Java API already contains a rich interface called Collection, which describes a bunch of useful operations on collections, described in an earlier lecture. Here we ll use our own much simplified version, called MyCollection. 1 interface MyCollection { 2 public boolean contains(comparable item); 3 public void add(comparable item); 4 public void remove(comparable item); 5 } The methods allow one to see if the collection contains a data item and to add or remove a data item. The data items we use must be objects from a class which implements the standard Comparable interface; notice how we can use the interface name as an object type. Stripped of comments, the Comparable interface looks like this: 1 interface Comparable { 2 public int compareto(object o); 3 } It specifies the compareto method: given two objects d1 and d2, we can compare them by testing the sign of d1.compareto(d2), as we showed for String in Note 17. We now begin a slightly different version of the BinTree class given in Lecture Note 24. The new class is SearchTree. It starts like this: 1 class SearchTree implements MyCollection { 2 static class Node { 3 Node left; // left subtree 4 Comparable data; // label we can use compareto on 5 Node right; // right subtree 6 7 public Node(Comparable d) { // Construct a leaf 8 data = d; 9 } 10 } 11 private Node root; 2
3 Line 1 contains the phrase implements MyCollection which is a claim that the methods of the MyCollection interface will be provided by the class. The Java compiler will check this requirement is met, and give an error message if not. On line 2, a static inner class called Node is declared. The static modifier makes Node behave much like an ordinary toplevel class, but one which is only available locally inside SearchTree. Compared with BinTree, we now have an extra layer: an instance of SearchTree has an object reference root to the root of the tree, which may be empty (null). The tree representation is then built using instances of Node. The reason for adding this extra layer is that the empty tree is now a proper object in the SearchTree class, rather than null as it was for BinTree. This is rather better programming style, because it follows principles of data abstraction: the exact representation of the tree should be hidden from the user of the class. It s also more convenient: the user can create a tree object with new SearchTree(), using the default constructor of SearchTree), and add items to the tree incrementally. The contains method To see if the tree contains a given item, we first compare the item with the root of the tree. If it is equal, we have found the item and finish. If it is smaller, we recursively search in the lefthand tree. If it is bigger, we recursively search in the righthand tree. If we hit an empty tree, we have failed. To implement this algorithm, we use two contains methods. The user calls the public instance method giving the item to be found. This calls a private helper method which is where the recursion happens. 14 // Is item in the tree? 15 public boolean contains(comparable item) { 16 return contains(root, item); 17 } 18 private static boolean contains(node t, Comparable item) { 19 if (t == null) { 20 // Object not found in empty tree 21 return false; 22 } else { 23 int cmp = t.data.compareto(item); 24 if (cmp == 0) { 25 // found the item 26 return true; 27 } else if (cmp>0) { 28 // data is greater than item, search left tree 29 return contains(t.left, item); 30 } else { 31 // data is less than item, search right tree 32 return contains(t.right, item); 33 } 34 } 35 } As shown in Lecture Note 24, an alternative (more objectoriented) way to implement 3
4 the recursion is to use a single instance method instead of the static helper method, so that the first recursive call above would read left.contains(item). Notice however that we would need to place this alternative method inside the Node class, since it recurses on Node objects. You may like to try rephrasing the SearchTree class following this pattern. The add method The add method is based on the same recursion as the contains method. We compare the item to be added with nodes in the tree, tracing a path left or right accordingly. If we hit an equal item, we do nothing the tree already contains the item. (A set collection does not record number of repetitions). Otherwise, we eventually come to a leaf position which is a proper place to add the new item. We implement this in the same way, using a public instance method and a private static helper method. The recursive method returns a Node object which represents the argument tree t, modified to ensure that item is included. 37 // Add an item to the tree 38 public void add(comparable item) { 39 root = add(root, item); 40 } 41 private static Node add(node t, Comparable item) { 42 if (t == null) { 43 // The (sub)tree is empty, make a new leaf node 44 return new Node(item); 45 } else { 46 int cmp = t.data.compareto(item); 47 if (cmp == 0) { 48 // data is already in the tree, no change 49 return t; 50 } else if (cmp>0) { 51 // data is greater than item, add in left tree 52 t.left = add(t.left, item); 53 return t; 54 } else { 55 // data is less than item, add in right tree 56 t.right = add(t.right, item); 57 return t; 58 } 59 } 60 } 4
5 The remove method The remove operation is a bit more complicated. Removing a leaf from the tree is easy enough, we simply set the parent s left or right reference to null. Removing a node which has a single subtree is also easy: we replace the node with the root of the subtree. The difficulty comes when we need to remove an internal node which has two subtrees: we need to combine the two subtrees into the place where one was. A possible approach is to define a merge operation which merges two binary trees, and use that to combine the two subtrees. We will follow another approach here, which does not actually remove an internal node in the tree; instead it removes a leaf node, and moves the label from the leaf node to the internal node. First of all, we define the remove operation following the same recursive scheme as before, but invoking an auxiliary method deletenode when we find the item to be deleted. 62 // Remove an item from the tree 63 public void remove(comparable item) { 64 root = remove(root, item); 65 } 66 private static Node remove(node t, Comparable item) { 67 if (t == null) { 68 // Item not found; do nothing 69 return null; 70 } else { 71 int cmp = t.data.compareto(item); 72 if (cmp == 0) { 73 // found the item here, let s delete this node 74 return deletenode(t); 75 } else if (cmp>0) { 76 // data greater than item, remove from left tree 77 t.left = remove(t.left, item); 78 return t; 79 } else { 80 // data less than item, remove from right tree 81 t.right = remove(t.right, item); 82 return t; 83 } 84 } 85 } Now we will define the operation to delete a node t. Things are easy for leaf nodes and nodes with only one child: if either subtree of t is empty, we can simply replace t by the other subtree. For an interior node with two children, what we do is find a suitable new label for the node from the labels already in the tree. If you think about this, there are two possible candidates: the greatest element in the left subtree, or the least element in the right subtree. Here we choose the latter. 5
6 87 // Delete this node from the tree. 88 // Should only be called on nonnull nodes 89 private static Node deletenode(node t) { 90 if (t.left == null) { 91 // Replace with right subtree 92 t = t.right; 93 } else if (t.right == null) { 94 // Replace with left subtree 95 t = t.left; 96 } else { 97 // Find least node in right subtree, 98 Node least = findleftmost(t.right); 99 // Copy its label here 100 t.data = least.data; 101 // Delete the least node in the right subtree. 102 t.right = deleteleftmost(t.right); 103 } 104 return t; 105 } Two operations support the rearrangement of the leaf node. First, findleftmost locates the left most node in a tree by chasing the left branch until there is no subtree. By the binary tree property, this leftmost node must be the least element in the tree. Second, deleteleftmost deletes the left most node in a tree, using a similar recursion. 106 // Find leftmost node. 107 // Should only be called on nonnull nodes 108 private static Node findleftmost(node t) { 109 if (t.left!= null) { 110 return findleftmost(t.left); 111 } else { 112 return t; 113 } 114 } 115 // Delete leftmost node. 116 // Should only be called on nonnull nodes 117 private static Node deleteleftmost(node t) { 118 if (t.left!= null) { 119 t.left = deleteleftmost(t.left); 120 return t; 121 } else { 122 // This is the leftmost node, replace it 123 // with its right subtree. 124 return t.right; 125 } 126 } 6
7 Testing The fragments above complete the implementation of the basic operations for binary trees. For testing purposes, it s useful to implement a tostring() method which shows the structure of the tree. It uses parenthesis for subtrees and to separate the node label. 128 // String representation of tree 129 public String tostring() { 130 return "<Binary search tree: " + tostring(root) + ">"; 131 } 132 private static String tostring(node tree) { 133 if (tree!= null) { 134 return ("(" tostring(tree.left) " " + tree.data + " " tostring(tree.right) + ")"); 138 } else { 139 return ""; 140 } 141 } We can test the code using a main method like the one given for binary trees. You can find some basic test code online in the file SearchTreeTest.java, which shows the example tree given on page Efficiency of binary search tree operations The binary search tree provides a reasonably fast implementation of a set. What does this mean, precisely? To compare efficiencies (time or space) of different algorithms, Computer Scientists use the BigO notation already mentioned in Lecture Note 23. We ll describe the meaning of this notation and algorithm analysis more exactly in later lectures. For now, it is enough to have some intuition that asserting a method that takes O(1) time means that it runs in some constant amount of time, whereas O(n) time means that it runs in some time proportional to n, where n is some measure of the size of the input. Generally, O(f(n)) means proportional to f(n). We call O(n) algorithms linear (think of the linear search in an array) and O(log n) algorithms logarithmic (e.g. binary search in an ordered array takes logarithmic time: why?). Let s analyse our tree search tree methods to try to understand how their run times vary depending on the input. First, notice that each of the operations contains, add, and remove makes a number of recursive calls equal to the length of the path in the tree that is followed. If findleftmost and deleteleftmost are called, this path must include the route to the smallest element of the right subtree. Under some simplifying assumptions, we can suppose that each of the methods contains, add and remove takes O(1) time (i.e., some constant amount) plus the time for the recursive call. Thus, if the distance from the root to any of the nodes concerned is h, the running time of each of the methods is O(h). Thus, the running time of the 7
8 methods is at most proportional to the height of the tree. But what is the height of a typical binary search tree of n nodes? Worst Case In the worst case, all the nodes in the binary tree will be arranged in a single path, like the right branch of the tree on page 1. This kind of tree would result if we took a sorted (or reverse sorted) list of elements and added them one at a time into an empty tree. There are also zigzag trees in which the single path does not consist of only right (or only left) children, but is a mixture of left and right children with the path taking a left or right turn at each interior node. The height of a nnode tree with one of these degenerate forms is n 1. We thus expect that contains, add and remove will take O(n) time on a collection of n elements if the representation of that collection happens to be one of these unfortunate trees. On average the element we want will be halfway down requiring n/2 nodes to be considered. Since each operation requires first searching for the element, each operation takes O(n) for one of these bad trees. Best Case A binary tree need not grow long and thin; instead it can be short and full, more like the first example on page 1. When every interior node down to some level has both children present and the next level has all leaves is called a full or complete tree. If we added the element 3 to the first example tree, it would become full. A full binary tree of height h has 2 h+1 1 nodes. This can be restated as: a binary tree of n = 2 h+1 1 nodes has a height of h. If we take logarithms we have log 2 (n+1) = h+1 or approximately, the height of the tree, h, is O(log n). Since the runtime of contains, add and remove is proportional to the height of the tree, we can conclude that on a binary tree that these operations take time that is logarithmic in the number of nodes. This performance is much better than the linear time taken for the long and thin tree in the worst case. As the set grows larger the running time of the tree operations grows much more slowly. Typical Case Is the worst case or the best case the more common case? It depends on how our trees are constructed, of course, and whether we make any attempt to balance them. If we make no such attempt, and assume that the search tree is built from inserting a set of randomly ordered elements, then it turns out that the standard operations take logarithmic time, on average, for a typical tree. Although the proof of this is beyond this scope of CS1, the essential point is that the expected value of the length of a path from the root of such a tree to a random node is O(log n). David Aspinall, 2003/11/27 18:06:01. 8
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