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COSC2007 Data Structures II Chapter 10 Trees I

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2 Topics Terminology

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3 Introduction & Terminology Review Position-oriented ADT: Insert, delete, or ask questions about data items at specific position Examples: Stacks, Queues Value-oriented ADT: Insert, delete, or ask questions about data items containing a specific value

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4 Introduction & Terminology Hierarchical (Tree) structure: Parent-child relationship between elements of the structure Nonlinear One-to-many relationships among the elements Examples: Organization chart Library card-catalog Contents of Books More???

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5 Introduction & Terminology Corporate structure

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6 Introduction & Terminology Example: Library card catalog Card Catalog Subject Catalog Author CatalogTitle Catalog A - AbbexStafford, R - Stanford, RZon - Zz Stafford, R. H.Standish, T. A. Stanford Research Institute

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7 Introduction & Terminology Table of contents of a book:

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8 Trees and Tree Terminology A tree is a data structure that consists of a set of nodes and a set of edges (or branches) that connect nodes.

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9 Introduction & Terminology A is the root node. B is the parent of D and E. C is the sibling of B D and E are the children of B D, E, F, G, I are external nodes, or leaves A, B, C, H are internal nodes The depth (level) of E is 2 The height of the tree is 3/4 The degree of node B is 2

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10 Introduction & Terminology Path from node n 1 to n k : A sequence of nodes n 1, n 2, …. n k such that there is an edge between each pair of nodes (n 1, n 2 ) (n 2, n 3 ),... (n k-1, n k ) Height h: The number of nodes on the longest path from the root to a leaf Ancestor-descendent relationship Generalization of parent-child relationship Ancestor of node n: A node on the path from the root to n Descendent of node n: A node on the path from n to a leaf

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11 Introduction & Terminology Nodes: Contains information of an element Each node may contain one or more pointers to other nodes A node can have more than one immediate successor (child) the lies directly below it Each node has at most one predecessor (parent) the lies directly above it

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12 Binary Trees A binary tree is an ordered tree in which every node has at most two children. A binary tree is: either empty or consists of a root node (internal node) a left binary tree and a right binary tree Each node has at most two children Left child Right child

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13 Binary Trees Special trees Left (right) subtree of node n: In a binary tree, the left (right) child (if any) of node n plus its descendants Empty BT: A binary tree with no nodes

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14 Number Of Nodes- height is h Minimum number of nodes in a binary tree: h At least one node at each of first h levels. Minimum number of nodes in a binary tree: 2 h - 1 All possible nodes at first h levels are present

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15 Number Of Nodes & Height Let n be the number of nodes in a binary tree whose height is h. h <= n <= 2 h – 1 log 2 (n+1) <= h <= n

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16 A B DE H IJK C FG L MNO Full Binary Trees In a full binary tree, every leaf node has the same depth every non-leaf node (internal node) has two children. A B DE H IJK C FG L Not a full binary tree.A full binary tree.

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17 Complete Binary Tree In a complete binary tree every level except the deepest must contain as many nodes as possible ( that is, the tree that remains after the deepest level is removed must be full). at the deepest level, all nodes are as far to the left as possible. A B DE H IJK C FG L Not a Complete Binary Tree A B DE H IJK C FG L A Complete Binary Tree

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18 Proper Binary Tree In a proper binary tree, each node has exactly zero or two children each internal node has exactly two children. A B DE H IJK C FG L A B DE H IJK C FG Not a proper binary treeA proper binary tree

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19 An Aside: A Representation for Complete Binary Trees Since tree is full, it maps nicely onto an array representation. A B DE H IJK C FG L A B C D E F G H I J K L T: last

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20 Properties of the Array Representation Data from the root node is always in T[0]. Suppose some node appears in T[i] data for its parent is always at location T[(i-1)/2] (using integer division) data for its children nodes appears in locations T[2*i+1] for the left child T[2*i+2] for the right child formulas provide an implicit representation of the edges can use these formulas to implement efficient algorithms for traversing the tree and moving around in various ways.

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21 Not Complete or Full or Proper A B D E H IJK C F L

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22 Balanced vs. Unbalanced Trees 1 A B DE H IJK C FG L root A B D E H I JK C FG L start Sort of Balanced Mostly Unbalanced A binary tree in which the left and right subtrees of any node have heights that differ by at most 1

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23 Binary search tree (BST) A binary tree where The value in any node n is greater than the value in every node in n s left subtree The value in any node n is less than the value of every node in n's right subtree The subtrees are binary search trees too

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24 Comparing Trees These two trees are not the same tree! A B A B Why?

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