Tree Definitions

• Trees give us a way to maintain O(lg n) access time on large amounts of data.
• They also give us a way to maintain a parent-child relationship among data.
• We will do an overview of trees
• Some Terms:
  • A tree is a collection on nodes, one of which is the root and all others are members of sub trees of this root. A tree can be empty.
  • The connection between any two nodes in a tree is called an edge.
  • An edge runs from a parent to a child node.
  • A node with no children is known as a leaf node.
  • All children of the same node ar known as sibling
  • A path from node n₁ to n_k is defined as a sequence of nodes n₁,n₂...n_k such that n_i is the parent of n_i+1
  • The length of a path is the number of edges in the path.
    • The path from a parent to a child is of length 1
    • Each node has a path to it's self of length 0
  • The depth of node n_i is the length of a path from the root to this node.
    • The root has depth 0
    • All children of the root have depth 1
  • The height of a node is the length of the longest path from that node to a leaf node.
  • If there is a path from n₁ to n₂ then n₁ is an ancestor of n₂ and n₂ is a descendant of n₁
  • If the path from n₁ to n₂ has length of 1 or more, then the relationship is a proper one, or n₁ is an proper ancestor of n₂
• Some tree applications
  • A family tree. Each node represents a preson.
    • The relationship between nodes is a real parent-child relationship
    • It doesn't make sense to delete a node in the tree.
    • Order among siblings is important.
    • Inserting is done at any node?
  • A file system can be stored as a tree.
    • Each reagular file is a leaf node.
    • Each internal node is a directory.
    • Empty directories are leaf nodes as well.
    • Only leaf nodes may be deleted.
    • Order among siblings is not important.
    • An insert can be performed at any node.
  • A binary search tree is an ADT where the binary search tree property must be followed.
    • BST Property
      • If x is a node in a BST, then
      • If y is in the right subtree of x, then x < y
      • If y is in the left subtree of x, then x > y
    • The relationship among the siblings is defined by the BST property.
    • It makes sense to delte any node, but if it is a non-leaf node it must be replaced by one of its children.
    • An insert is only performed at the leaf nodes.
  • Notice insert, delete are different depending on the application.
• ADT Tree:
  • Domain: homogenious data where there is a parent-child relationship among the data.
  • Operations:
    • root(), returns an iterator to the root node of a tree.
    • element(i), returns the element at iterator i
    • parent(i), child(i), returns an iterator to parent/child
      • error for root or leaf nodes respectively
    • Children accessors:
      • children(i) - iterator to traverse children
      • firstChild(i) - iterator to first child of i
      • rightSibling(i) - iterator to right sibling of i
      • leftSibling(i) -
    • IsInternal(), IsLeaf(), IsRoot()
    • depth(v), height(v),
    • Others depending on the implementation.
      • Insert, delete
    • elements() - return an iterator over all elements
    • positions() - returns an iterator over all positions.