3.2   Binary Search Trees



We examine a symbol-table implementation that combines the flexibility of insertion in linked lists with the efficiency of search in an ordered array. Specifically, using two links per node leads to an efficient symbol-table implementation based on the binary search tree data structure, which qualifies as one of the most fundamental algorithms in computer science.

Definition. A binary search tree (BST) is a binary tree where each node has a Comparable key (and an associated value) and satisfies the restriction that the key in any node is larger than the keys in all nodes in that node's left subtree and smaller than the keys in all nodes in that node's right subtree.

Anatomy of a binary tree                Anatomy of a binary search tree


Basic implementation.

Program BST.java implements the ordered symbol-table API using a binary search tree. We define a inner private class to define nodes in BST. Each node contains a key, a value, a left link, a right link, and a node count. The left link points to a BST for items with smaller keys, and the right link points to a BST for items with larger keys. The instance variable N gives the node count in the subtree rooted at the node. This field facilitates the implementation of various ordered symbol-table operations, as you will see.

Subtree counts in a BST

Analysis.

The running times of algorithms on binary search trees depend on the shapes of the trees, which, in turn, depends on the order in which keys are inserted.

Best case for BST                Typical case for BST                Worst case for BST
It is reasonable, for many applications, to use the following simple model: We assume that the keys are (uniformly) random, or, equivalently, that they are inserted in random order.

Proposition.

Search hits in a BST built from N random keys requires ~ 2 ln N (about 1.39 lg N) compares on the average.

Proposition.

Insertion and search misses in a BST built from N random keys requires ~ 2 ln N (about 1.39 lg N) compares on the average.

The visualization below shows the result of inserting 255 keys in a BST in random order. It displays the number of keys (N), the maximum number of nodes on a path from the root to a leaf (max), the average number of nodes on a path from the root to a leaf (avg), the average number of nodes on a path from the root to a leaf in a perfectly balanced BST (opt).

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Order-based methods and deletion.

An important reason that BSTs are widely used is that they allow us to keep the keys in order. As such, they can serve as the basis for implementing the numerous methods in our ordered symbol-table API.

Proposition.

Search, insertion, finding the minimum, finding the maximum, floor, ceiling, rank, select, delete the minimum, delete the maximum, delete, and range count operations all take time proportional to the height of the tree, in the worst case.


Exercises

  1. Give five orderings of the keys A X C S E R H that, when inserted into an initially empty BST, produce the best-case tree.

    Solution. Any sequence that inserts H first; C before A and E; S before R and X.

  2. Add to BST.java a method height() that computes the height of the tree. Develop two implementations: a recursive method (which takes linear time and space proportional to the height), and method like size() that adds a field to each node in the tree (and takes linear space and constant time per query).
  3. Write a test client TestBST.java for use in testing the implementations of min(), max(), floor(), ceiling(), select(), rank(), deleteMin(), deleteMax(), and keys() that are given in the text.
  4. Give nonrecursive implementations of get(), put(), and keys() for BST.

    Solution: NonrecursiveBST.java

Creative Problems

  1. Perfect balance. Write a program PerfectBalance.java that inserts a set of keys into an initially empty BST such that the tree produced is equivalent to binary search, in the sense that the sequence of compares done in the search for any key in the BST is the same as the sequence of compares used by binary search for the same set of keys.

    Hint: Put the median at the root and recursively build the left and right subtree.

  2. Certification. Write a method isBST() in BST.java that takes a Node as argument and returns true if the argument node is the root of a binary search tree, false otherwise.
  3. Subtree count check. Write a recursive method isSizeConsistent() in BST.java that takes a Node as argument and returns true if the subtree count field N is consistent in the data structure rooted at that node, false otherwise.
  4. Select/rank check. Write a method isRankConsistent() in BST.java that checks, for all i from 0 to size() - 1, whether i is equal to rank(select(i)) and, for all keys in the BST, whether key is equal to select(rank(key)).

Web Exercises

  1. The great tree-list recursion problem. A binary search tree and a circular doubly linked list are conceptually built from the same type of nodes - a data field and two references to other nodes. Given a binary search tree, rearrange the references so that it becomes a circular doubly-linked list (in sorted order). Nick Parlante describes this as one of the neatest recursive pointer problems ever devised. Hint: create a circularly linked list A from the left subtree, a circularly linked list B from the right subtree, and make the root a one node circularly linked list. Them merge the three lists.
  2. BST reconstruction. Given the preorder traversal of a BST (not including null nodes), reconstruct the tree.
  3. True or false. Given a BST, let x be a leaf node, and let y be its parent. Then either (i) the key of y is the smallest key in the BST larger than the key of x or (ii) the key of y is the largest key in the BST smaller than the key of x. Answer: true.
  4. True or false. Let x be a BST node. The next largest key (successor of x) can be found by traversing up the tree toward the root until encountering a node with a non-empty right subtree (possibly x itself); then finding the minimum key in the right subtree (by following its rightmost path).
  5. Tree traversal with constant extra memory. Describe how to perform an inorder tree traversal with constant extra memory (e.g., no function call stack).

    Hint: on the way down the tree, make the child node point back to the parent (and reverse it on the way up the tree).

  6. Reverse a BST. Given a standard BST (where each key is greater than the keys in its left subtree and smaller than the keys in its right subtree), design a linear-time algorithm to transform it into a reverese BST (where each key is smaller than the keys in its left subtree and greater than the keys in its right subtree). The resulting tree shape should be symmetric to the original one.
  7. Level-order traversal reconstruction of a BST. Given a sequence of keys, design a linear-time algorithm to determine whether it is the level-order traversal of some BST (and construct the BST itself).
  8. Find two swapped keys in a BST. Given a BST in which two keys in two nodes have been swapped, find the two keys.

    Solution. Consider the inorder traversal a[] of the BST. There are two cases to consider. Suppose there is only one index p such that a[p] > a[p+1]. Then swap the keys a[p] and a[p+1]. Otherwise, there are two indices p and q such a[p] > a[p+1] and a[q] > a[q+1]. Let's assume p < q. Then, swap the keys a[p] and a[q+1].