Trees

1
Trees
2
General out tree

root

siblings children of
the same node
subtree
internal nodes
or nonleaves
external nodes, or leaves


3
Binary Tree

• Binary tree recursive definition.
• An empty tree is a binary tree.
• If L and R are binary trees then one node called
  a root joined with L and R is also a binary tree.
• Depth or level of a node distance from the
  root.
• Root has depth 0. Any other node has distance
  that equals 1 plus distance of its parent.
• Height of a tree maximal depth of all leaves.
• Height of a node is height of a subtree where
  this node is a root.

4
Definitions

• Degree of a node number of children for that
  node
• Internal nodes nodes with at least one child.
• Internal nodes have positive degree
• External nodes nodes with no children.
• External nodes have degree zero.

5
Definitions

• Complete Binary Tree is a binary tree in which
  all internal nodes have degree 2 and all leaves
  are at the same hepth.

6
Lemmas

• There are at most 2d nodes at depth d of a binary
  tree.
• A binary tree with height h has at most
• 2 h1 1 nodes.
• A binary tree with n nodes has height at least
  ceiling lg(n1) -1.

7
Binary TreeTraversal

• void traverse (BinTree T)
• if ( T is not empty)
• 1
• traverse (leftSubtree(T))
• 2
• traverse (rightSubtree(T))
• 3
• process root(T) is either in point 1 or in
  point 2 or in point 3 respectively to get
  preorder, inorder or postorder traversal.

8
General trees

• A General out-tree is a nonempty structure with
  nodes and directed edges, such that one node
  called a root has no incoming edges and all other
  nodes have exactly one incoming edge.
• A forest is a collection of separate trees.
• Every node in a tree is a root of its own
  subtree.
• In - tree has edges oriented from child toward
  the parent instead of away from it as in out tree.

9
Definitions

• Tree rooted at w is called a principal subtree of
  the tree rooted at v
• v
• w

10
Principal Subtree Properties

• Each principal subtree has fewer nodes than the
  whole tree.
• Subtrees do not have to be ordered for general
  trees.
• There is a set of subtrees for every node of a
  general tree.
• Subtrees in a binary tree are ordered.
• There is left and right subtree. This is a
  sequence of two subtrees.

11
General Tree ADT

• Tree buildTree( object newRoot, TreeList
  oldTrees)
• Preconditions none
• Postconditions If xbuildTree( object newRoot,
  TreeList oldTrees) then
• X refers to a newly created object
• root(x) newRoot
• Subtrees(x) oldTrees

12
General Tree ADT

• Assume that subtrees are ordered.
• Object root(Tree t)
• PreconditionNone
• TreeList subtrees (tree t)
• PreconditionNone

13
TreeList ADT

• TreeList ADT is the analog of IntList with class
  Tree in place of int for the element type.
• TreeList cons (Tree t, TreeList rSiblings)
• Tree first (TreeList siblings)
• TreeList rest(TreeList siblings)
• TreeList nil

14
General Tree Traversal

• Page 85 Fig 2.13