Trees

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Trees
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Definition

• A general tree set of nodes and edges that
  connect them.
• Exactly one path between 2 nodes
• Path connected sequence of edges

• A rooted tree
• One distinguished node is called the root
• Every node c, except root, has one parent p, the
  first node on path from c to the root
• Root has no parent and a node can have any number
  of children

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Terminology

• Leaf node with no children
• Sibling nodes with same parent
• Ancestors of a node A nodes on path from d to
  root, including A (?), As parent, As
  grandparent,, root.
• If A is ancestor of D, then D is a descendant of
  A
• Length of path number of edges (NOT vertices) in
  the path
• Depth of node N length of path from N to root
• Depth of root is zero
• Height of node N length of path from N to its
  deepest descendant
• Height of any leaf is zero
• Height of a tree height of the root
• Subtree rooted at N tree formed by N and its
  descendants.
• Size number of nodes

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Examples

• Leaf C, D, F, H
• Sibling B and G, C, D, and E
• Height of the tree
• Size of the tree
• Depth of H
• Ancestors of node D

• What happens if there is a link between D and F?
• Is it a tree?
• Why?

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Another example

• The structure shown on the right was extracted
  from Gene Ontology. Round rectangles represent
  nodes (GO terms) and arrows stand for edges
  indicating the relationship between two nodes.
• Is it a tree structure? Why?

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Tree traversal

• Tree traversal a manner of visiting each node in
  a tree once.
• Visiting a node performing an action (such as
  printing, examining, updating,) on a node.
• Always starting from the root
• Four ways
• Preorder traversal
• Postorder traversal
• Inorder traversal
• Level order traversal

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Pre-order traversal

• Definition Visit each node before recursively
  visiting its children left to right.
• Root visited first
• Each node visited only once, so a pre-order
  traversal takes O(n) times, where n is the number
  of nodes in a tree.
• Examples
• Printing out the file directory
• in a natural way

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Post-order Traversal

• Visiting each nodes children (from left to
  right) before the node itself.
• Visit the root last
• To do a post-order traversal of a general tree
• Do a post-order traversal each of the subtrees of
  the root one-by-one in the order given and then
• visit the root.
• Examples
• Expression tree
• Calculation of total disk space
• taken by a file directory

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Level-order traversal

• Visit the root (depth zero), then all the nodes
  at depth one, then all the nodes at depth two,
  and so on.
• At each depth the nodes are visited from left to
  right
• Non recursive traversal
• Queue-based implementation
• Enqueue the root
• while (queue is not empty)
• Dequeue a node
• Visist it
• Enqueue its children from left to right
• Until queue is empty

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In-order traversal

• Inorder traversal only makes sense for binary
  trees
• Traverse left child, and then
• Visit node, and then
• Traverse right child
• Recursive implementation

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Trees types and examples

• Types of tree
• A general tree
• A tree has no limit to the number of children
• Binary tree
• Each node has at most 2 children
• Balanced tree
• If all the leaves of the tree are on the same
• level or at least within one level of each other
• AVL trees, binary search tree
• Everyday life
• A family tree
• The organization of sport tournaments
• Computer applications
• A file system
• A web page
• Expression tree the parent nodes are the
  operators, and the children are the operands

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Exercises

• Given the following tree

A
C
B
G
D
E
F
I
H

• List the nodes of the above general tree in the
  following order
• Pre-order
• Post-order
• Level-order