Lecture 23, Fall 2018 Monday October 29

Transcription

1 Binary search trees Oliver W. Layton CS231: Data Structures and Algorithms Lecture 23, Fall 2018 Monday October 29

2 Plan Tree traversals Binary search trees Code up binary tree

3 Pre-order traversal 1. Process the root. 2. Process the nodes in the le1 subtree with a recursive call 3. Process the nodes in the right subtree with a recursive call Let's see what visi-ng the nodes on the board with a pre-order traversal would look like.

4 In-order traversal 1. Process the nodes in the le0 subtree with a recursive call 2. Process the root. 3. Process the nodes in the right subtree with a recursive call Let's see what visi-ng the nodes on the board with a in-order traversal would look like. How does in-order traversal get its name?

5 In-order traversal pseudocode (1/2) Because binary trees are recursively defined (why?), the code for pre-order and in-order traversals are concise. Let's assume: Each node holds ints The tree isn't empty We get current node's children by the instance variables left and right. We inifate the traversal by calling this recursive funcfon on the root node.

6 In-order traversal pseudocode (2/2) public void inorderprint() { if (left!= null) left.inorderprint(); System.out.println(data + " "); if (right!= null) right.inorderprint(); }

7 Post-order traversal 1. Process the nodes in the le0 subtree with a recursive call 2. Process the nodes in the right subtree with a recursive call 3. Process the root. Post-order gets its name from the fact that we postpone processing the root un6l we first deal with le9 and right subtrees. Let's see what visi6ng the nodes on the board with a in-order traversal would look like.

8 Post-order traversal pseudocode public void postorderprint() { if (left!= null) left.postorderprint(); if (right!= null) right.postorderprint(); System.out.println(data + " "); }

9 Binary Search Trees (BSTs) In-order traversal of the example binary tree produced the integers 1-9 in sorted order. The tree is special and is called a binary search tree. Exci?ng news: we finally have a data structure that can let us find stuff faster than linear search! Let's see why the example tree is a BST.

10 Binary Search Trees: Defini1on (Preliminary) A binary search tree (BST) is a tree where: 1. The LEFT subtree has nodes with 'numbers' that are LESS THAN the parent. 2. The RIGHT subtree has nodes with 'numbers' that are GREATER THAN the parent. Is the defini+on recursive (are subtrees valid BSTs?)?

11 Advantage of binary search trees (1/2) BSTs can be substan.ally faster to search for items than a linear search (for loop through linear array). What's the height of a full binary tree? This means that on average searching for items is only! Why?

12 Advantage of binary search trees (2/2)

13 Key-value pairs (1/2) Binary search trees generally store more general data than just ints. They o7en store data as key-value pairs. Think of a BST node as a look-up table: It maps keys to values that we store. Key/value pairs are also called entries.

14 Examples: Key-value pairs Address book: Key Value Bob Karen Sue

15 Key-value pairs (2/2) O#en, we organize the BST according to the keys. Once we find the key in the tree (name), we look up the associated value (phone number). Key tend to be compact and helpful for looking up the value --- usually Integers or Strings. Unique keys map to unique values (i.e. duplicate keys return the same value). The value could be a single piece of data (a string, number, etc), arrays, or even whole data structures.

16 Example Generic KeyValuePair class class KeyValuePair<K, V> { private K key; private V value; } public KeyValuePair(K key, V value) { this.key = key; this.value = value; } public K getkey() { return key; } public V getvalue() { return value; } public void setvalue(v value) { this.value = value; }

17 Binary Search Trees: Defini1on (Updated) A binary search tree (BST) is a tree where: 1. The LEFT subtree has nodes with KEYS that are LESS THAN the parent. 2. The RIGHT subtree has nodes with KEYS that are GREATER THAN the parent.

18 Binary Search Trees: Implementa4on Let's implement a simple BST (Integer rather than KeyValuePair). How should we compare data stored in two separate nodes?