Chapter 6. Trees I

6.1 The Tree Abstract Data Type

• In computer science, a tree is an abstract model of a hierarchical structure
• A tree consists of nodes with a parent-child relation
• Applications:
• Organization charts
• File systems
• Programming environments
• Subtree: tree consisting of a node and its descendants

• A tree T is a set of nodes storing elements in a parent-child relationship with the following properties:
  • T has a special node r, called the root of T, with no parent node.
  • Each node v of T different from r has a unique parent node u.
• If node u is the parent of node v, we say that v is a child of u.
• Two nodes that are children of the same parent are siblings.
• A node is external if has no children, and it is internal if it has one or more children. External nodes are also known as leaves.
• The subtree of T rooted at a node v is the tree consisting of all the descendents of v in T (including v itself).
• An ancestor of a node is either the node itself or an ancestor of the parent of the node.
• A node v is a descendent of a node u if u is an ancestor of v.
• Example:

Root: node without parent (A) Internal node: node with at least one child (A, B, C, F) External node (a.k.a. leaf ): node without children (E, I, J, K, G, H, D) Ancestors of a node: parent, grandparent, grand-grandparent, etc. Depth of a node: number of ancestors Height of a tree: maximum depth of any node (3) Descendant of a node: child, grandchild, grand-grandchild, etc.

• A tree is ordered if there is a linear ordering defined for children of each node (siblings ordering).
• A binary tree is an ordered tree in which every node has at most two children.
• A binary tree is proper if each node has either zero or two children.
• For each internal node in a binary tree, we label each child as either being a left child or a right child.
• The subtree rooted at a left or right child of an internal node v is called a left subtree or right subtree of v, respectively.

• We use positions to abstract nodes
• Generic methods:
• integer size()
• boolean isEmpty()
• ObjectIterator elements()
• PositionIterator positions()
• Accessor methods:
• Position root()
• Position parent(Position)
• PositionIterator children(Position)
• Query methods:
• boolean isInternal(Position)
• boolean isExternal(Position)
• boolean isRoot(Position)
• Update methods:
• void swapElements(Position, Position)
• Object replaceElement(Position, Object)
• Additional update methods may be defined by data structures implementing the Tree ADT

6.2 Basic Algorithms on Trees

• If v is an external node, then the height of v is 0.
• Otherwise, the height of v is one plus the maximum height of a child of v.