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: path length from Depth Graph E) G= : ( " ' nodes finite set of vertices root . :* :: e.ci :* :i . " is ( pairs of vertices ) IF . - acyclic graph " graph " Directed


  1. ⇒ : path length from Depth Graph E) G= : ( ÷÷÷÷÷÷÷" ' nodes finite set of vertices root . :* :÷÷÷÷÷÷÷÷÷:÷¥ e.ci :* :i . " is ( pairs of vertices ) IF . - acyclic graph " graph " Directed ← TO %) ° Degree ( of node ) : number ↳ to-do 0 OT , off 't ? ?aegs4 8¥40 of children ftp.g?o:ningw7.68doDesreedkogtrtiee' " o_0 " " digraph ianmgxiode why ' § Rooted " " " To ene .de root 4731 Formal definition : " Relations EDM " " is either Family Rooted tree : - single node ( root ) ta - grandparent 041010610 go a- ' - set of one or more - - - - - ' - parent . ? too . be 0/10 - rooted trees ( : " subtrees " ) ¥ . " " . leaf : . " i. joined to a common root ' : I ' ' i. o ' a. child ( / ) . . 0 . ' sibling 's leaf ; noch ; ¥nb ← - grandchild . "

  2. ⇐ ⇐ wasted space ? rooted trees Representing : : A binary tree with Theorem root Each node stores a ( linked ) Eg " . list of its children n nodes has links htt null ③ ⑤ ' ① t.jo?jn%4f.s96 0h Node structure : ' ① ⑤ → next sibling Ff onneodmewe ' " " " " " " first child root £ : " class Btnodegejjodhatra FT In Java : %75inraeenteaetiinmT.in?.tnodese → 7¥d data ; Trees E BTNodehE > left ; > right %Q : ⑧ % lmk why w L u n root : ④ t ⑧ ( not full ) Full : ⇒ go a f " # # called the Binary representation ' ⑦ ⑥ ① ④ ① ⑤ Ye ⑤ Binary tree :D rooted tree t " ' i :& ' " ④ Of ③ 1$ ⑥ of degree 2 , where each " ' Node structure : ' ' ÷ n' Yun two children node has ( possibly null ) lefts right data : Every - leaf node Full non " £7 left ! right has 2 children

  3. m complete Binary Tree : All levels traverse ( Btnodev ) l Traversals How to ( systematically ) visit : full ( except last ) null ) return ; of a rooted tree ? if ( v the nodes ( can be generalized ) visitors - Preorder 09-0 Binary Tree Traversals 2isirsproes@fiIInorderroot.i.gov - process In Innings ;¥÷i÷ Eiichi . . .de . , 4hq" - Lily parental } Challenge : traversals dy so ÷÷÷÷÷÷÷:÷ensimmM TO *d ④ / " n u , _ - a In order : An extended binary atbO*# µ DThm : § ④ tree with n internal nodes Another way to save space . . . Those wasteful null links Threaded binary tree : ( black ) has nil external . . " " " . . . store ( useful ) links in the nodes ( blue ) Q Extended binary ' O null links .( Use a mark bit O %§gQg : Replace IB to distinguish link types Observation : Every tree . ) extended binary tree each null link with E. g. In order Threads : a special leaf node :b BID I 'D Null left - inorder predecessor is full external node Null rights " successor 4

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