Hierarchical Collections I

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Hierarchical Collections I

• Introduction to Trees

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What Is a Tree?

• Each item has at most one predecessor
• Each item can have many successors

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What Is a Tree?
Root node

• Start with a single node, called the root

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What Is a Tree?
Root node

• Start with a single node, called the root
• Each successor node is a child or daughter node

Daughters of root node
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What Is a Tree?
Root node

• Successors are also called descendants

Descendants of root node
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What Is a Tree?
Root node

• Nodes without successors are called leaf nodes
• The set of leaf nodes is the frontier

Leaf nodes (the frontier)
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What Is a Tree?
Root node

• Nodes with at least one successor are called
  interior nodes

Interior nodes
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What Is a Tree?
Root node

• Predecessors are also called ancestors
• The immediate predecessor is called the parent

Ancestors of node X
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What Is a Tree?
Root node

• Nodes with the same parent are called siblings

Siblings
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What Is a Tree?
Root node

• Levels in a tree are numbered from 0

Level 0
Level 1
Level 2
Level 3
There are three nodes in level 3
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What Is a Tree?
Root node

• A binary tree allows at most two successors per
  node

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What Is a Tree?
Root node

• A general tree allows any number of successors
  per node

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Tree Applications

• File directories
• Processing sentences (computer programs or
  natural languages)
• Searchable data structures
• Heaps (implement heap sort, priority queues)

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A Parse (Expression) Tree
Source language program
The value of the expression
Ok or not OK
A parse tree
Syntax error messages
Semantic error messages
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Trees for Binary Expressions

• An expression tree for a number is a node
  containing the number
• Otherwise, the tree is a node containing an
  operator and links to left and right subtrees
• The subtrees contain the operands of the
  expression

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Example Expression Trees
23
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Example Expression Trees
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Root node Right subtree Left subtree
3 5
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Example Expression Trees
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Root node Right subtree Left subtree
3 5
3 5 4
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Example Expression Trees
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Root node Right subtree Left subtree
3 5
(3 5) 4
3 5 4
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Operator Precedence
(3 5) 4
3 5 4
Operators with higher precedence appear lower in
the tree, unless overridden by parentheses.
5 4 3
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Operator Precedence
5 4 - 3
5 4 - 3 6
5 4 - 3
When operators have equal precedence, the ones to
the left appear lower in the tree.