Trees

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Topic 12

• Trees

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Chapter Objectives

• define trees as data structures
• define the terms associated with trees
• discuss the possible implementations of trees
• analyze tree implementations of collections
• discuss methods for traversing trees
• examine a binary tree example

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Trees

• a tree is a nonlinear data structure used to
  represent things that are in some hierarchical
  relationship, for example
• family tree
• class inheritance hierarchy in Java
• computer file system (directories and
  subdirectories)
• decision trees
• top-down design

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Tree Definition

• tree a set of elements of the same type that is
  either empty or has a distinguished element
  called the root, from which descend zero or more
  subtrees
• each subtree is itself a tree

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Tree Terminology

• nodes the elements in the tree
• a node has a data portion and some reference
  portions
• nodes are connected by edges
• root the origin of the tree
• leaf node a node without a reference to another
  node
• interior node a node that is not a leaf node

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Tree Terminology
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Tree Terminology (contd)

• a tree has no cycles (i.e. a node cant refer
  back to a node that is higher in the hierarchy)
• the simplest tree is the empty tree
• what does that mean? the root reference is null

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Tree Terminology (contd)

• parent or predecessor the node directly above in
  the hierarchy
• child or successor a node directly below in the
  hierarchy
• siblings nodes that have the same parent
• example diagram

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Discussion

• Does a leaf node have any children?
• Does the root node have a parent?
• How many parents does every node other than the
  root node have?

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Tree Terminology (contd)

• Continuing the family tree analogy
• ancestors of a node s are the parent of s, the
  parent of the parent of s, etc.
• descendants of a node t are the children of t,
  the children of the children of t, etc.

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Level of a Node

• level of a node number of edges between root
  and node
• it can be defined recursively
• the root node is at level 1
• a node that is not the root node has a level that
  is 1 greater than the level of its parent
• example

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Path length and level
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Height of a Tree

• height of a tree is the number of nodes on the
  longest path from the root to a leaf
• alternate definition
• is a tree is empty, its height is 0
• is a tree is not empty, its height is the maximum
  of the levels of its nodes
• example

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Subtrees

• subtree a set of nodes which is itself a tree
• a subtree consists of a node and all its
  descendants
• example
• c is the root node of a subtree

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Balanced and Complete Trees

• a tree is balanced if all the leaves are on the
  same level or at least within one level of each
  other
• a tree is complete if it is balanced and all the
  leaves at the bottom level are on the left side
  of the tree

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Balanced and unbalanced trees
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Some complete trees
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Degree or Arity

• the degree or arity of a node is the number of
  its children
• example
• the degree or arity of a tree is the maximum of
  the degrees of its nodes

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Binary Trees

• general tree a node may have any number of
  children
• n-ary tree a node may have no more than n
  children
• binary tree a node may have no more than 2
  children
• example

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Binary Trees (contd)

• each node of a binary tree will have
• a data portion
• a reference to a left child
• a reference to a right child
• recursive definition of a binary tree
• a binary tree is the empty tree, or a binary tree
  whose left and right successors are binary trees
• a binary tree is a positional tree, i.e. it
  matters whether the subtree is left or right

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Tree Traversals

• a traversal of a tree should visit each node of
  the tree once
• example a typical reason to traverse a tree is
  to display the data stored at each node of the
  tree
• traversal orderings
• preorder
• inorder
• postorder
• level-order

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Preorder Traversal

• start at the root
• visit each node, followed by its children
• recursive algorithm
• if tree is not empty,
• visit root node of tree
• perform preorder traversal of its left subtree
• perform preorder traversal of its right subtree

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Inorder Traversal

• start at the root
• visit the left child of each node, then the node,
  then any remaining nodes
• recursive algorithm
• if tree is not empty,
• perform inorder traversal of left subtree of root
• visit root node of tree
• perform preorder traversal of its right subtree

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Postorder Traversal

• start at the root
• visit the children of each node, then the node
• recursive algorithm
• if tree is not empty,
• perform postorder traversal of left subtree of
  root
• visit root node of tree
• perform postorder traversal of right subtree of
  root

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Traversal Analysis

• each node is processed exactly once
• so, a traversal is O(n)

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Iterative Traversals

• (do we want to do this??)
• recursive tree traversals use the Java runtime
  stack to keep track of where we are in the tree
• in iterative tree traversals, it is up to the
  programmer!
• etc. etc. etc.

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Operations on a Binary Tree

• what might we want to do with a binary tree?
• add an element (but where?)
• remove an element (but from where?)
• is the tree empty?
• size of the tree (how many elements)
• other operations?

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Discussion

• difficult to have a general add operation, until
  we know the purpose of the tree (we will discuss
  binary search trees later)
• could add randomly go either right or left,
  and add at the first available spot

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Discussion (contd)

• similarly, where would a general remove operation
  remove from?
• could arbitrarily choose to remove, say, the
  leftmost leaf
• if random choice, what would happen to the
  children and descendants of the element that was
  removed? what does the parent of the removed
  element now point to?
• what if the removed element is the root?

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One Set of Operations on a Binary Tree
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UML description of the BinaryTreeADT interface
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Binary Tree Implementation using Links

• to represent the binary tree, we need
• a linked list of nodes
• root to keep track of the node that is the root
  of the tree
• count to keep track of the number of nodes in
  the tree
• first, how will we represent a node?

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BinaryTreeNode class

• this represents a node in a binary tree and
  contains
• the data element stored in the node
• pointer to left child of the node
• pointer to right child of the node
• see BinaryTreeNode.java

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BinaryTree class

• it implements the Binary Tree ADT interface (see
  BinaryTreeADT.java)
• constructors
• removeLeftSubtree method
• find method
• iteratorInOrder method
• see others in BinaryTree.java

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Using Binary Trees Expression Trees

• expression tree represents an arithmetic
  expression
• root and internal nodes contain operations
• leaf nodes contain operands
• we can use an expression tree to evaluate an
  expression evaluate from the bottom up

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An example expression tree
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Building Expression Trees

• how do we build an expression tree?
• recall stack algorithm to evaluate postfix
  expressions
• exercise develop the algorithm

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Building an expression tree from a postfix
expression
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Building an expression tree from a postfix
expression (contd)
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UML description of the Postfix example