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Trees

Trees. Overview. Tree Representation and Terminology Binary Trees Binary Search Trees Pointer-Based Representation of a Binary Tree Array-Based Representation of a Binary Tree. Tree. A tree is a collection of nodes and directed edges , satisfying the following properties:

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Trees

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  1. Trees

  2. Overview • Tree • Representation and Terminology • Binary Trees • Binary Search Trees • Pointer-Based Representation of a Binary Tree • Array-Based Representation of a Binary Tree

  3. Tree • A tree is a collection of nodes and directed edges, satisfying the following properties: • There is one specially designated node called the root, which has no edges pointing to it. • Every node except the root has exactly one edge pointing to it. • There is a unique path (of nodes and edges) from the root to each node.

  4. Trees: NotTrees: root edge node GraphicalRepresentation • Trees, as defined on the preceding slide, are typically drawn with circles (or rectangles) representing the nodes and arrows representing the edges. • The root is typically placed at the top of the diagram, with the rest of the tree below it. Leaf nodes

  5. Terminology (Cont’d.) • If an edge goes from node a to node b, then a is called the parent of b, and b is called a child of a. • Children of the same parent are called siblings. • If there is a path from a to b, then a is called an ancestor of b, and b is called a descendent of a. • A node with all of its descendants is called a subtree. • If a node has no children, then it is called a leaf of the tree. • If a node has no parent (there will be exactly one of these), then it is the root of the tree.

  6. A B C F D E G H I J K Terminology: Example • A is the root • D, E, G, H, J & K are leaves • B is the parent of D, E & F • D, E & F are siblings and children of B • I, J & K are descendants of B • A & B are ancestors of I subtree

  7. (These two binary trees are distinct.) Binary Trees • Intuitively, a binary tree is LIKE a tree in which each node has no more than two children. • Formally, a binary tree is a set T of nodes such that either: • T is empty, or • T consists of a single node, r, called the root, and two (non-overlapping) binary trees, called the left and rightsubtrees of r. • UNLIKE trees, binary trees may be empty, and they distinguish between left and rightsubtrees.

  8. Data Data Data Data Data Data Data Left_child Left_child null null null Left_child null Right_child null null null Right_child Right_child null Sample Binary Tree Root

  9. 21 John 3 34 Brenda Peter 55 2 8 Amy Mary Tom 5 13 Binary Search Trees • A binary search tree is a binary tree in which each node, n, has a value satisfying the following properties: • n’s value is > all values in its left subtree, TLeft, • n’s value is < all values in its right subtree, TRight, and • TLeftand TRightare both binary search trees.

  10. Terminology (Cont’d.) • Intuitively, the level of a node is the number of nodes on a path from the root to the node. • Formally, level of node, n: • If n is the root of a tree, then it is at level 1. • Otherwise, its level is 1 greater than the level of its parent. • Height of binary tree, T: • If T is empty, its height is 0. • Otherwise, its height is the maximum level of its nodes, or, equivalently, 1 greater than the height of the root’s taller subtree. Namely, height(T) = 1 + max { height( TLeft ), height(TRight ) }

  11. Terminology (Cont’d.) • Intuitively, a binary tree is full if it has no missing nodes. • Formally, a binary tree of height h is fullif • It is empty (h = 0). • Otherwise, the root’s subtrees are full binary trees of height h – 1. • If not empty, each node has2 children, except the nodes at level h which have no children. • Alternatively, the binary tree has all of its leaves at level h.

  12. Terminology (Cont’d.) • Intuitively, a binary tree of height h is complete if it is full down to level h – 1, and level h is filled from left to right. • Formally, a binary tree of height h is complete if • All nodes at level h – 2 and above have 2 children each, • If a node at level h – 1 has children, all nodes to its left at the same level have 2 children each, and • If a node at level h – 1 has 1 child, it is a left child.

  13. Terminology (Cont’d.) • A binary tree is balanced if the difference in height between any node’s left and right subtree is  1. • Note that: • A full binary tree is also complete. • A complete binary tree is not always full. • Full and complete binary trees are also balanced. • Balanced binary trees are not always full or complete.

  14. Operations on Tree • Different operations on a Tree are as follows • Insertion • Deletion • Searching • Traversal • Pre-Order • In-Order • Post- Order • Please follow the Power Point Presentations on Insertion, Deletion and Traversal using Binary Search Tree.

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