Trees

1
Trees
2
Definition of a tree

• A tree is a node with a value and zero or more
  children
• Depending on the needs of the program, the
  children may or may not be ordered

• 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)

3
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

4
Data structure for a tree

• 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 TreeltVgt V value ArrayList
  children
• If we dont care about the order of children, we
  might use a Set instead of a ArrayList

5
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

6
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

7
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

8
Part of a genealogy
Elaine
Pauline
Chester
Eugene
Paula
Carol
David
Winfred
Danielle
Steven
Isaac
9
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

10
Trees for expressions and statements

• Examples

11
More trees for statements

• while (n gt 1) exp x exp n--

• for (int i 0 i lt n i) ai 0

12
The End