# 6_trees - Binary Search Trees ! Understand ! Understand...

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Binary Search Trees

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! Understand tree terminology ! Understand and implement tree traversals ! Define the binary search tree property ! Implement binary search trees ! Implement the TreeSort algorithm October 2004 John Edgar 2

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! A set of nodes (or vertices) with a single starting point ! called the root ! Each node is connected by an edge to another node ! A tree is a connected graph ! There is a path to every node in the tree ! A tree has one fewer edge than the number of nodes October 2004 John Edgar 4
October 2004 John Edgar 5 yes! NO! All the nodes are not connected NO! There is an extra edge (5 nodes and 5 edges) yes! (but not a binary tree) yes! (it’s actually the same graph as the blue one)

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A B C D G E F ! Node v is said to be a child of u , and u the parent of v if ! There is an edge between the nodes u and v , and ! u is above v in the tree, ! This relationship can be generalized ! E and F are descendants of A ! D and A are ancestors of G ! B, C and D are siblings ! F and G are? October 2004 John Edgar 6 root edge parent of B, C, D
! A leaf is a node with no children ! A path is a sequence of nodes v 1 v n ! where v i is a parent of v i +1 (1 i n-1 ) ! A subtree is any node in the tree along with all of its descendants ! A binary tree is a tree with at most two children per node ! The children are referred to as left and right ! We can also refer to left and right subtrees October 2004 John Edgar 7

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October 2004 John Edgar 8 C A B C D G E F leaves : C,E,F,G path from A to D to G subtree rooted at B
October 2004 John Edgar 9 A B C G D E left subtree of A H I J F right subtree of C right child of A

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! The height of a node v is the length of the longest path from v to a leaf ! The height of the tree is the height of the root ! The depth of a node v is the length of the path from v to the root ! This is also referred to as the level of a node ! Note that there is a slightly different formulation of the height of a tree ! Where the height of a tree is said to be the number of different levels of nodes in the tree (including the root) October 2004 John Edgar 10
October 2004 John Edgar 11 A B C G D E H I J F height of node B is ? height of the tree is ? depth of node E is ? level 1 level 2 level 3 2 3 2

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October 2004 John Edgar 13 yes! yes! However, these trees are not “beautiful” (for some applications)

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! A binary tree is perfect , if ! No node has only one child ! And all the leaves have the same depth ! A perfect binary tree of height h has how many nodes? ! 2 h +1 – 1 nodes, of which 2 h are leaves October 2004 John Edgar 14 A B C G D E F
October 2004 John Edgar 15 12 22 31 23 24 33 34 35 36 38 01 11 21 32 37 " Each level doubles the number of nodes " Level 1 has 2 nodes (2 1 ) " Level 2 has 4 nodes (2 2 ) or 2 times the number in Level 1 " Therefore a tree with h levels has 2 h+1 - 1nodes " The root level has 1 node the bottom level has 2 h nodes, that is, just over ½ the nodes are leaves

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! A binary tree is
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## This note was uploaded on 04/17/2010 for the course CMPT 11151 taught by Professor Gregorymori during the Spring '10 term at Simon Fraser.

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6_trees - Binary Search Trees ! Understand ! Understand...

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