Trees

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Trees
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Definition of a tree

• A tree is like a binary tree, except that a node
  may have any number of children
• Depending on the needs of the program, the
  children may or may not be ordered

• Like a binary tree, a tree has a root, internal
  nodes, and leaves
• Each node contains an element and has branches
  leading to other nodes (its children)
• Each node (other than the root) has a parent
• Each node has a depth (distance from the root)

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More definitions

• An empty tree has no nodes
• The descendents of a node are its children and
  the descendents of its children
• The ancestors of a node are its parent (if any)
  and the ancestors of its parent
• The subtree rooted at a node consists of the
  given node and all its descendents
• An ordered tree is one in which the order of the
  children is important an unordered tree is one
  in which the children of a node can be thought of
  as a set
• The branching factor of a node is the number of
  children it has
• The branching factor of a tree is the average
  branching factor of its nodes

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Data structure for a tree

• A node in a binary tree can be represented as
  follows
• class BinaryTreeNode Object value
  BinaryTreeNode leftChild, rightChild
• However, each node in a tree has an arbitrary
  number of children, so we need something that
  will hold an arbitrary number of nodes, such as
  an ArrayList
• class TreeNode Object value ArrayList
  children
• If we dont care about the order of children, we
  might use a Set instead of a ArrayList

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ADT for a tree

• It must be possible to
• Construct a new tree
• If a tree can be empty, this may require a header
  node
• Add a child to a node
• Get (iterate through) the children of a node
• Access (get and set) the value in a node
• It should probably be possible to
• Remove a child (and the subtree rooted at that
  child)
• Get the parent of a node

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Traversing a tree

• You can traverse a tree in preorder
• void preorderPrint(node) System.out.println(
  node) Iterator iter node.children.iterator(
  ) while (iter.hasNext())
  preorderPrint(iter.next())
• You can traverse a tree in postorder
• void postorderPrint(node) Iterator iter
  node.children.iterator() while
  (iter.hasNext()) postorderPrint(iter.nex
  t()) System.out.println(node)
• You cant usually traverse a tree in inorder
• Why not?

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Other tree manipulations

• Theres really nothing new to talk about youve
  seen it all with binary trees
• A tree consists of nodes, each node has
  references to some other nodesyou know how to do
  all this stuff
• There are some useful algorithms for searching
  trees, and with some modifications they also
  apply to searching graphs
• Lets move on to some applications of trees

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File systems

• File systems are almost always implemented as a
  tree structure
• The nodes in the tree are of (at least) two
  types folders (or directories), and plain files
• A folder typically has childrensubfolders and
  plain files
• A folder also contains a link to its parentin
  both Windows and UNIX, this link is denoted by ..
• In UNIX, the root of the tree is denoted by /
• A plain file is typically a leaf

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Family trees

• It turns out that a tree is not a good way to
  represent a family tree
• Every child has two parents, a mother and a
  father
• Parents frequently remarry
• An upside down binary tree almost works
• Since it is a biological fact (so far) that every
  child has exactly two parents, we can use left
  child father and right child mother
• The terminology gets a bit confusing
• If you could go back far enough, it becomes a
  mathematical certainty that the mother and father
  have some ancestors in common

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Part of a genealogy
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Game trees

• Trees are used heavily in implementing games,
  particularly board games
• A node represents a position on the board
• The children of a node represent all the possible
  moves from that position
• More precisely, the branches from a node
  represent the possible moves the children
  represent the new positions
• Planning ahead (in a game) means choosing a path
  through the tree
• However
• You cant have a cycle in a tree
• If you can return to a previous position in a
  game, you have a cycle
• Graphs can have cycles

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Binary trees for expressions

• Ordered trees can be used to represent arithmetic
  expressions

• To evaluate an expression (given as a node)
• If it is a leaf, the element in it specifies the
  value
• If the element is a number, that number is the
  value
• If the element is a variable, look up its value
  in a table
• If it is not a leaf, the element in it specifies
  an operation
• Evaluate the children and combine them according
  to the operation

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(General) trees for expressions

• You can use binary trees for expressions if you
  have only unary and binary operators
• Java has a ternary operator

• Trees can be used to represent statements as well
  as expressions
• Statements can be evaluated as easily as
  expressions

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More trees for statements

• while (n 1) exp x exp n--

• for (int i 0 i

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Writing compilers and interpreters

• A compiler does three things
• Parses the input program (converts it into an
  abstract syntax tree)
• (Optionally) optimizes the abstract syntax tree
• Traverses the tree and outputs assembly language
  or machine code to do the same operations
• An interpreter does three things
• Parses the input program (converts it into an
  abstract syntax tree)
• (Optionally) optimizes the abstract syntax tree
• Traverses the tree in an order controlled by the
  node contents, and performs the operations as it
  goes
• Parsing is usually the hard part, but there is a
  very simple technique (called recursive descent
  parsing) that can be used if the language is
  carefully designed and you dont care too much
  about efficiency or good error messages

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Ill never need to write a compiler...

• Are you sure?
• If you cant parse text inputs, you are limited
  to reading simple things like numbers and Strings
• If you can parse text input, you can make sense
  of
• tell Mary "Meet me at noon"
• fire phasers at 3, 7
• 17.25, 0.203 8.97i, 0.95i
• 2812"48'
• 330pm-5pm
• Parsing is less important in these days of GUIs,
  but its still pretty important
• Java provides basic support for parsing with its
  StringTokenizer and StreamTokenizer classes

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The End