Trees CMSC 202 Version 502 Tree Basics 1

Trees CMSC 202, Version 5/02

Tree Basics 1. A tree is a set of nodes. 2. A tree may be empty (i. e. , contain no nodes). 3. If not empty, there is a distinguished node, r, called the root and zero or more non-empty subtrees T 1, T 2, T 3, …. Tk, each of whose roots is connected by a directed edge from r. 4. Trees are recursive in their definition and, therefore, in their implementation. CMSC 202, Version 5/02 2

A General Tree A B C G D E H F CMSC 202, Version 5/02 I K J L

Tree Terminology • Tree terminology takes its terms from both nature and genealogy. • A node directly below the root, r, of a subtree is a child of r, and r is called its parent. • All children with the same parent are called siblings. • A node with one or more children is called an internal node. • A node with no children is called a leaf or external node. CMSC 202, Version 5/02 4

Tree Terminology (con’t) • A path in a tree is a sequence of nodes, (N 1, N 2, … Nk) such that Ni is the parent of Ni+1 for 1 <= i <= k. • The length of this path is the number of edges encountered (k – 1). • If there is a path from node N 1 to N 2, then N 1 is an ancestor of N 2 and N 2 is a descendant of N 1. CMSC 202, Version 5/02 5

Tree Terminology (con’t) • The depth of a node is the length of the path from the root to the node. • The height of a node is the length of the longest path from the node to a leaf. • The depth of a tree is the depth of its deepest leaf. • The height of a tree is the height of the root. • True or False – The height of a tree and the depth of a tree always have the same value. CMSC 202, Version 5/02 6

Tree Storage • First attempt - each tree node contains – The data being stored • We assume that the objects contained in the nodes support all necessary operations required by the tree. – Links to all of its children • Problem: A tree node can have an indeterminate number of children. So how many links do we define in the node? CMSC 202, Version 5/02 7

First Child, Next Sibling • Since we can’t know how many children a node can have, we can’t create a static data structure -- we need a dynamic one. • Each node will contain – The data which supports all necessary operations – A link to its first child – A link to a sibling CMSC 202, Version 5/02 8

First Child, Next Sibling Representation • To be supplied in class CMSC 202, Version 5/02 9

Tree Traversal • Traversing a tree means starting at the root and visiting each node in the tree in some orderly fashion. “visit” is a generic term that means “perform whatever operation is applicable”. • “Visiting” might mean – Print data stored in the tree – Check for a particular data item – Almost anything CMSC 202, Version 5/02 10

Breadth-First Tree Traversals • • Start at the root. Visit all the root’s children. Then visit all the root’s grand-children. Then visit all the roots great-grandchildren, and so on. • This traversal goes down by levels. • A queue can be used to implement this algorithm. CMSC 202, Version 5/02 11

BF Traversal Pseudocode Create a queue, Q, to hold tree nodes Q. enqueue (the root) while (the queue is not empty) Node N = Q. dequeue( ) for each child, X, of N Q. enqueue (X) • The order in which the nodes are dequeued is the BF traversal order. CMSC 202, Version 5/02 12

Depth-First Traversal • • • Start at the root. Choose a child to visit; remember those not chosen Visit all of that child’s children, and so on. Repeat until all paths have been traversed • This traversal goes down a path until the end, then comes back and does the next path. • A stack can be used to implement this algorithm. CMSC 202, Version 5/02 13

DF Traversal Pseudocode Create a stack, S, to hold tree nodes S. push (the root) While (the stack is not empty) Node N = S. pop ( ) for each child, X, of N S. push (X) The order in which the nodes are popped is the DF traversal order. CMSC 202, Version 5/02 14

Binary Trees • A binary tree is a tree in which each node may have at most two children and the children are designated as left and right. • A full binary tree is one in which each node has either two children or is a leaf. • A perfect binary tree is a full binary tree in which all leaves are at the same level. CMSC 202, Version 5/02 15

A Binary Tree CMSC 202, Version 5/02

A binary tree? A full binary tree? CMSC 202, Version 5/02 17

A binary tree? A full binary tree? A perfect binary tree? CMSC 202, Version 5/02 18

Binary Tree Traversals • Because nodes in binary trees have at most two children (left and right), we can write specialized versions of DF traversal. These are called – In-order traversal – Pre-order traversal – Post-order traversal CMSC 202, Version 5/02 19

In-Order Traversal • At each node – visit my left child first – visit me – visit my right child last 8 5 9 15 CMSC 202, Version 5/02 3 12 6 7 10 2 20

In-Order Traversal Code void in. Order. Traversal(Node *node. Ptr) { if (node. Ptr != NULL) { in. Order. Traversal(node. Ptr->left. Ptr); cout << node. Ptr->data << endl; in. Order. Traversal(node. Ptr->right. Ptr); } } CMSC 202, Version 5/02

Pre-Order Traversal • At each node – visit me first – visit my left child next – visit my right child last 8 5 9 15 CMSC 202, Version 5/02 3 12 6 7 10 2 22

Pre-Order Traversal Code void pre. Order. Traversal(Node *node. Ptr) { if (node. Ptr != NULL) { cout << node. Ptr->data << endl; pre. Order. Traversal(node. Ptr->left. Ptr); pre. Order. Traversal(node. Ptr->right. Ptr); } } CMSC 202, Version 5/02

Post-Order Traversal • At each node – visit my left child first – visit my right child next – visit me last 8 5 9 15 CMSC 202, Version 5/02 3 12 6 7 10 2 24

Post-Order Traversal Code void post. Order. Traversal(Node *node. Ptr) { if (node. Ptr != NULL) { post. Order. Traversal(node. Ptr->left. Ptr); post. Order. Traversal(node. Ptr->right. Ptr); cout << node. Ptr->data << endl; } } CMSC 202, Version 5/02

Binary Tree Operations • Recall that the data stored in the nodes supports all necessary operators. We’ll refer to it as a “value” for our examples. • Typical operations: – Create an empty tree – Insert a new value – Search for a value – Remove a value – Destroy the tree CMSC 202, Version 5/02 26

Creating an Empty Tree • Set the pointer to the root node equal to NULL. CMSC 202, Version 5/02 27

Inserting a New Value • The first value goes in the root node. • What about the second value? • What about subsequent values? • Since the tree has no properties which dictate where the values should be stored, we are at liberty to choose our own algorithm for storing the data. CMSC 202, Version 5/02 28

Searching for a Value • Since there is no rhyme or reason to where the values are stored, we must search the entire tree using a BF or DF traversal. CMSC 202, Version 5/02 29

Removing a Value • Once again, since the values are not stored in any special way, we have lots of choices. Example: – First, find the value via BF or DF traversal. – Second, replace it with one of its descendants (if there any). CMSC 202, Version 5/02 30

Destroying the Tree • We have to be careful of the order in which nodes are destroyed (deallocated). • We have to destroy the children first, and the parent last (because the parent points to the children). • Which traversal (in-order, pre-order, or post-order) would be best for this algorithm? CMSC 202, Version 5/02 31

Giving Order to a Binary Tree • Binary trees can be made more useful if we dictate the manner in which values are stored. • When selecting where to insert new values, we could follow this rule: – “left is less” – “right is more” • Note that this assumes no duplicate nodes (i. e. , data). CMSC 202, Version 5/02 32

Binary Search Trees • A binary tree with the additional property that at each node, – the value in the node’s left child is smaller than the value in the node itself, and – the value in the node’s right child is larger than the value in the node itself. CMSC 202, Version 5/02 33

A Binary Search Tree 50 57 42 30 22 CMSC 202, Version 5/02 53 34 67

Searching a BST • Searching for the value X, given a pointer to the root 1. If the value in the root matches, we’re done. 2. If X is smaller than the value in the root, look in the root’s left subtree. 3. If X is larger than the value in the root, look in the root’s right subtree. • A recursive routine – what’s the base case? CMSC 202, Version 5/02 35

Inserting a Value in a BST • To insert value X in a BST 1. Proceed as if searching for X. 2. When the search fails, create a new node to contain X and attach it to the tree at that node. CMSC 202, Version 5/02 36

Inserting 100 38 56 150 20 40 125 138 90 CMSC 202, Version 5/02

Removing a Value From a BST • Non-trivial • Three separate cases: – node is a leaf (has no children) – node has a single child – node has two children CMSC 202, Version 5/02 38

Removing 100 150 50 70 30 20 40 60 55 53 CMSC 202, Version 5/02 130 80 65 57 120 85

Destroying a BST • The fact that the values in the nodes are in a special order doesn’t help. • We still have to destroy each child before destroying the parent. • Which traversal must we use? CMSC 202, Version 5/02 40