COSC2007 Data Structures II

Chapter 10

Trees I

2

Topics

Terminology

3

Introduction & Terminology

Review Position-oriented ADT:

Insert, delete, or ask questions about data items at specific position

Examples: Stacks, Queues Value-oriented ADT:

Insert, delete, or ask questions about data items containing a specific value

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Introduction & Terminology

Hierarchical (Tree) structure: Parent-child relationship between elements of the

structure Nonlinear One-to-many relationships among the elements

Examples: Organization chart Library card-catalog Contents of Books More???

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Introduction & Terminology Corporate structure

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Introduction & Terminology

Example: Library card catalogCard

Catalog

Subject Catalog

Author Catalog

Title Catalog

A - Abbex Stafford , R - Stanford, R

Zon - Zz

Stafford, R. H.

Standish, T. A.

Stanford Research Institute

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Introduction & Terminology Table of contents of a book:

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Trees and Tree Terminology

A tree is a data structure that consists of a set of nodes and a set of edges (or branches) that connect nodes.

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Introduction & Terminology A is the root node. B is the parent of D and E. C is the sibling of B D and E are the children of B D, E, F, G, I are external nodes, or

leaves A, B, C, H are internal nodes The depth (level) of E is 2 The height of the tree is 3/4 The degree of node B is 2

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Introduction & Terminology Path from node n1 to nk:

A sequence of nodes n1, n2, … . nk such that there is an edge between each pair of nodes (n1, n2) (n2, n3), . . . (nk-1, nk)

Height h: The number of nodes on the longest path from the root

to a leaf Ancestor-descendent relationship

Generalization of parent-child relationship Ancestor of node n:

A node on the path from the root to n Descendent of node n:

A node on the path from n to a leaf

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Introduction & Terminology

Nodes: Contains information of an element Each node may contain one or more pointers

to other nodes A node can have more than one immediate

successor (child) the lies directly below it Each node has at most one predecessor

(parent) the lies directly above it

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Binary Trees A binary tree is an ordered tree in which every node has at most two children. A binary tree is:

either empty or consists of a

root node (internal node) a left binary tree and a right binary tree

Each node has at most two children Left child Right child

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Binary Trees

Special trees Left (right) subtree of node n:

In a binary tree, the left (right) child (if any) of node n plus its descendants

Empty BT: A binary tree with no nodes

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Number Of Nodes-height is h Minimum number of nodes in a binary tree: h

At least one node at each of first h levels. Minimum number of nodes in a binary tree: 2h - 1

All possible nodes at first h levels are present

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Number Of Nodes & Height

Let n be the number of nodes in a binary tree whose height is h.

h <= n <= 2h – 1

log2(n+1) <= h <= n

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A

B

D E

H I J K

C

F G

LM N O

Full Binary Trees In a full binary tree,

every leaf node has the same depth every non-leaf node (internal node) has two children.

A

B

D E

H I J K

C

F G

L

Not a full binary tree. A full binary tree.

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Complete Binary Tree In a complete binary tree

every level except the deepest must contain as many nodes as possible ( that is, the tree that remains after the deepest level is removed must be full).

at the deepest level, all nodes are as far to the left as possible.

A

B

D E

H I J K

C

F G

L

Not a Complete Binary Tree

A

B

D E

H I J K

C

F G

L

A Complete Binary Tree

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Proper Binary Tree In a proper binary tree,

each node has exactly zero or two children each internal node has exactly two children.

A

B

D E

H I J K

C

F G

L

A

B

D E

H I J K

C

F G

Not a proper binary tree A proper binary tree

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An Aside: A Representation for Complete Binary Trees

Since tree is full, it maps nicely onto an array representation.

A

B

D E

H I J K

C

F G

L

0 1 2 3 4 5 6 7 8 9 10 11 12

A B C D E F G H I J K LT:

last

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Properties of the Array Representation

Data from the root node is always in T[0]. Suppose some node appears in T[i]

data for its parent is always at location T[(i-1)/2]

(using integer division) data for it’s children nodes appears in locations

T[2*i+1] for the left child

T[2*i+2] for the right child formulas provide an implicit representation of the edges can use these formulas to implement efficient algorithms for

traversing the tree and moving around in various ways.

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Not Complete or Full or Proper

A

B

D E

H I J K

C

F

L

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Balanced vs. Unbalanced Trees 1

A

B

D E

H I J K

C

F G

L

root

A

B

D

E

H I

J K

C

F G

L

start

Sort of BalancedMostly Unbalanced

A binary tree in which the left and right subtrees of any node have heights that differ by at most 1

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Binary search tree (BST)

A binary tree where The value in any node n is greater than the

value in every node in n’s left subtree The value in any node n is less than the

value of every node in n's right subtree The subtrees are binary search trees too

60

7020

10 40

5030

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Comparing Trees

These two trees are not the same tree!

A

B

A

B

Why?