Tree ADT

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Tree ADT

• Definition
• A tree T is a finite set of one or more nodes
  such that there is one designated node r called
  the root of T, and the remaining nodes in (T
  r ) are partitioned into n ? 0 disjoint subsets
  T1, T2, ..., Tk, each of which is a tree, and
  whose roots r1 , r2 , ..., rk , respectively, are
  children of r.

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Tree ADT

• Characteristics
• A tree is an instance of a more general category
  called graph
• A tree consists of nodes connected by edges
• Typically there is one node in the top row of the
  tree, with lines connecting to more nodes on the
  second row, even more on the third, and so on.

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Tree ADT

• Root
• The node at the top of the tree is called the
  root.
• There is only one root in a tree.
• For a collection of nodes and edges to be defined
  a tree, there must be one (and only one!) path
  from the root to any other node .

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Tree ADT

• Root

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Tree ADT

• Path
• If n1, n2, ,nk is a set of nodes in the tree
  such that ni, is the parent of ni1 for 1? i ? k,
  then this set is called a path from n1 to nk. The
  length of the path is k 1.
• The number of edges that connect nodes is the
  length of the path.

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Tree ADT

• Parent
• Any node (except the root) has exactly one edge
  running upward to another node. The node above it
  is called the parent of the node.
• If there is a path from node R to node M, then R
  is an ancestor of M.
• Parents, grandparents, etc. are ancestors.

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Tree ADT

• Child
• The nodes below a given node are called its
  children.
• If there is a path from node R to node M, then M
  is a descendant of R.
• Children, grandchildren, etc. are descendants

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Tree ADT

• Leaf and Internal node
• A node that has no children is called a leaf node
  or simply a leaf. There can be only one root in a
  tree, but there can be many leaves.
• A node (apart from the root) that has children is
  an internal node.

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Tree ADT

• Levels
• The level of a particular node refers to how
  many generations the node is from the root. If
  the root is assumed to be on level 0, then its
  children will be on level 1, its grandchildren
  will be on level 2, and so on.

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Tree ADT

• Depth
• The depth of a node M in a tree is the length of
  the path from the root of the tree to M.
• Height
• The height of a tree is one more than the depth
  of the deepest node in the tree.

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Tree ADT

• Subtree
• Any node may be considered to be the root of the
  subtree, which consists of its children and its
  children children, and so on. If you think in
  terms of families, a nodes subtree contains all
  its descendants.

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Tree ADT

• Subtree

Subtree
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Tree ADT

• Visiting
• A node is visited when program control arrives
  at the node, usually for the purpose of carrying
  out some operation on the node, such as checking
  the value of one of its data fields, or
  displaying it.
• Merely passing over a node on the path form
  one node to another is not considered to be
  visiting the node.

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Tree ADT

• Traversing
• To traverse a tree means to visit all of the
  nodes in some specified order.
• For example, you might visit all the nodes in
  order of ascending key value. There are other
  ways to traverse a tree, as well see later.
• If traversal visits all of the tree nodes it is
  called enumeration.

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Tree ADT

• Keys
• One data item in an object is usually designated
  a key value. This value is used to search for the
  item or perform other operations on it.

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Tree ADT

• Binary Tree
• Made up of a finite set of nodes that is either
  empty or consists of a node called the root
  together with two binary trees, called the left
  and right subtrees, which are disjoint from each
  other and from the root.

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Tree ADT

• Full Binary Trees
• A binary tree where each node is either a leaf or
  is an internal node with exactly two non-empty
  children.
• That means, a node is allowed to have either none
  or two children (but not one!)

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Tree ADT

• Complete Binary Trees
• A binary tree whereby if the height is d, and all
  levels, except possibly level d, are completely
  full. If the bottom level is incomplete, then it
  has all nodes to the left side.
• That is the tree has been filled in the level
  order from left to right.

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Tree ADT
Not Complete Binary Tree
Complete Binary Tree