CSC 2300Data Structures & Algorithms

February 6, 2007

Chapter 4. Trees

Today – Trees

Tree basics Tree implementation Induction and trees Tree traversal Tree algorithm analysis Tree application: file system

Terminology

Rooted Tree

Recursive definition. A tree consists of a distinguished node r, called the root, and zero or more nonempty (sub)trees T1, T2, …, Tk, each of whose roots are connected by a directed edge from r.

Parent and child. The root r is the parent of each subtree root.

Tree

Definition of leaves? Which nodes are leaves? Definition of siblings? Which nodes are siblings? Grandparent and grandchild?

Path and Length

A path from node n1 to node nk is defined as a sequence of nodes n1, n2, …, nk such that ni is the parent of ni+1, for 1≤i<k.

The length of this path is the # of edges on the path, namely k-1. Is there a path of length zero? There is exactly one path from the root to each node.

Depth and Height

The depth of node ni is the length of the unique path from the root to ni. What is the depth of the root? The height of ni is the length of the longest path from ni to a leaf. What is the height of a leaf? The height of a tree is equal to the height of its root. The depth of a tree is equal to the depth of the deepest leaf. How is the height of a tree related to its depth?

Ancestor and Descendant

If there is a path from ni to nk, then ni is an ancestor of nk and nk is a descendant of ni.

If ni ≠ nk, then ni is a proper ancestor of nk and nk is a proper descendant of ni.

Implementation of Trees

Can you suggest an obvious way to implement a tree? Have in each node, besides its data, a link to each child of the node. Why is this approach possibly very inefficient? The # of children per node may vary greatly and is not known in advance. Is there a better solution?

Solution

Induction and Trees

Check Your Answer

m = 5, i = ?, l = ?

Check Your Answer

m = 2, i = ?, l = ?

Tree Traversals

Time Analysis of Tree Algorithm

Application

One popular use of tree is the directory structure in operating systems, including UNIX and DOS.

List Directory

Time Analysis

Line 1 is executed exactly once per node, since each name is output once. It follows that line 2 must also be executed once per node. Line 4 can be executed at most once for each child of each node. But the

number of children is exactly one less than the number of nodes. The “for” loop iterates once per iteration of line 4, plus each time the loop

ends. Thus, if there are N file names to be output, then the running time is O(N).

Traversal Strategy

What is the traversal strategy? In preorder traversal, work in a node is

performed before (pre) its children are processed.

Size of Each Directory

Suppose we know the number of disk blocks taken by each file. How can we find the size of each directory?

Postorder Traversal

In a postorder traversal, the work at a node is performed after (post) its children are evaluated.