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# lecture05 - Trees Trees Binary tree and Binary tree...

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1 Trees Trees Binary tree and Binary tree traversals Expression trees Infix, Postfix, Prefix notation Binary Search Trees Operations supported Complexity

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2 (Directed) Tree 7 6 3 4 3 2 1 8 7 1 6 9 2 3 9 8 2 8 0 5 4 6
3 Definition of Trees /\ is a tree. It is called an “empty tree”. If is a node and are non-empty trees where k 0, then is a tree. T 1 T 2 T k T 1 T 2 T k

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4 Notation about Trees root, subtrees parent, child, sibling leaves, internal nodes edge path, length ancestor, descendant depth, height Root at depth 0, One node tree : height 0 7 6 3 4 3 2 1 8 8 2 8
5 Tree criteria Always a single root A single path to every node Doesn’t matter how we draw it

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6 Trees ? 4
7 Game tree 1 2 7 2 1 3 8 6 X 1 2 7 2 1 X 8 6 3 1 2 7 2 1 3 8 X 6 Start state for the 8- puzzle; X marks the empty square. Can move the 3 tile into the empty square, or, can move the 6 tile into it. Goal of the game is to place all tiles in some special order.

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8 Binary Trees Binary tree : every node has at most 2 children (left child and right child) N-ary tree : no node has more than N children. Advantages? Disadvantages? Equivalencies? 7 6 5 4 3 2 1 8
9 Some Properties of Binary Trees Suppose a binary tree has n nodes, and height h. The total numbers of nodes: h+1 n 2 h+1 -1 The height of the tree: (from inequality above) log(n+1)-1 h (n-1)

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10 Implementation of Binary Trees 7 6 5 4 3 2 1 8 x 2 1 7 8 3 4 6 5
11 Binary Trees LL INFO RL node T L R (((a+b)*(c+(d-3))) LL: left link RL: Right Link * + + - d 3 a b c T

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12 Implementation of N-ary Trees 7 6 4 3 4 8 2 8 5 x
13 Implementation of Trees 7 6 4 3 4 8 2 8 4 7 8 3 6 4 2 8 x Right link to siblings; left link to successors

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14 Some applications Expressions ( (7+3*12+18) * (4/3) ) HTML or XML Word documents 3
15 Traversing a tree Traversing a tree is to visit every node in the tree

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