10/22/10
19:10:12 1
24
CS 61B: Lecture 24
Friday, October 22, 2010
Today’s reading: Goodrich & Tamassia, Chapter 7.
ROOTED TREES
============
A _tree_ consists of a set of nodes and a set of edges that connect pairs of
nodes. A tree has the property that there is exactly one path (no more, no
less) between any two nodes of the tree. A _path_ is a connected sequence of
zero or more edges.
In a _rooted_ tree, one distinguished node is called the _root_. Every node c,
except the root, has exactly one _parent_ node p, which is the first node
traversed on the path from c to the root. c is p’s _child_. The root has no
parent. A node can have any number of children.
Some other definitions:
- A _leaf_ is a node with no children.
- _Siblings_ are nodes with the same parent.
- The _ancestors_ of a node d are the nodes on the path from d to the root.
These include d’s parent, d’s parent’s parent, d’s parent’s parent’s
parent, and so forth up to the root. Technically, the ancestors of d also
include d itself, which makes you wonder about d’s sex life. The root is
an ancestor of every node in the tree.
- If a is an ancestor of d, then d is a _descendant_ of a.
- The _length_ of a path is the number of edges in the path.
- The _depth_ of a node n is the length of the path from n to the root. (The
depth of the root is zero.)
- The _height_ of a node n is the length of the path from n to its deepest
descendant. (The height of a leaf node is zero.)
- The height of a tree is the depth of its deepest node = height of the root.
- The _subtree_ rooted at node n is the tree formed by n and its descendants.
- A _binary_tree_ is a tree in which no node has more than two children, and
every child is either a _left_child_ or a _right_child_, even if it’s the
only child its parent has.
A commonly encountered application of trees is the directory structure of a
file system.
_______˜jrs/61b_______ <== Root node
/ | | \
/ | | \
hw index.html lab _lec__
/ \ /\ / /\ \ \_
/ \ ^ / \ / / \ \ \
hw1 hw2 | lab1 lab2 01 02 03 04 05 <== Leaf nodes
leaf node
Representing Rooted Trees
-------------------------
Goodrich and Tamassia present a data structure in which each node has three
references: one reference to an item, one reference to the node’s parent, and
one reference to the node’s children, which can be stored in any reasonable
data structure like a linked list. Directories are typically stored this way,
but the lists they use are represented very differently than our list ADTs.
Another popular tree representation spurns separately encapsulated linked lists
so that siblings are directly linked. It retains the "item" and "parent"
references, but instead of referencing a list of children, each node references
just its leftmost child. Each node also references its next sibling to the
right. The "nextSibling" references are used to join the children of a node in
a singly-linked list, whose head is the node’s "firstChild".
I’ll call this tree a "SibTree", since siblings are central to the
representation. The nodes are called "SibTreeNodes".
class SibTreeNode { | class SibTree {
Object item; | SibTreeNode root;
SibTreeNode parent; | int size;
SibTreeNode firstChild; | }
SibTreeNode nextSibling; |
} |
===============================================================================
+ ROOTED TREE | -------------------- ---------------------------- +
=============== |--- ---- | | parent | +
+ ||.|root size|14| | ---------------------------- +
+ |-+- ---- | | item | +
+ --|----------------- ---------------------------- +
+ v SibTree object | firstChild | nextSibling | +
+ ----- ---------------------------- +
+ | * | structure of SibTreeNodes +
+ ----- +
+ Root node => |jrs| +
+ -----<--------- +
+ |.|*| \ +
+ /----<---- \ +
+ / ^^ \ \ +
+ v / \ \ \ +
+ ---/- -\--- -\--- -\--- +
+ | . | | . | | . | | . | +
+ ----- ----- ----- ----- +
+ |hw | |ind| |lab| |lec|<------------------------ +
+ ----- ----- ----- -----<------------------ \ +
+ |.|.+->|*|.+->|.|.+->|.|*|<------------ \ \ +
+ /---- ----- /---- --\--<------ \ \ \ +
+ / ^^ / ^^ \ ^ \ \ \ \ +
+ v / \ v / \ \ \ \ \ \ \ +
+ ---/- -\--- ---/- -\--- >-\--- -\--- -\--- -\--- -\--- +
+ | . | | . | | . | | . | | . | | . | | . | | . | | . | +
+ ----- ----- ----- ----- ----- ----- ----- ----- ----- +
+ |hw1| |hw2| |lb1| |lb2| |01 | |02 | |03 | |04 | |05 | +
+ ----- ----- ----- ----- ----- ----- ----- ----- ----- +
+ |*|.+->|*|*| |*|.+->|*|*| |*|.+->|*|.+->|*|.+->|*|.+->|*|*| +
+ ----- ----- ----- ----- ----- ----- ----- ----- ----- +
===============================================================================
